# Explore

- Nate's ruminations
- Processes of Rationality
- An Effective Altruist's Guide to the Future
- Improve comments by tagging claims
- 43m
- A beginner's guide to explaining things
- AI alignment
- Accelerator Project
- Active learning for opaque, powerful predictors
- Alexei Andreev
- Alexei's Blog
- Algebraic field
- Ambitious vs. narrow value learning
- Anton Geraschenko
- Arbital
- Arbital Labs
- Arbital as single conversational locus
- Bone Fracture
- Calcium
- Cancer
- Cardiovascular Disease
- Central examples
- Claim explainer: donor lotteries and returns to scale
- Convex
- Crony belief
- Decision theory
- Early-life planning
- Effective altruism
- Eliezer Yudkowsky
- Eliezer office hours
- Eric Bruylant
- Geneva Macro Labs
- Geometric algebra
- Jaime Sevilla Molina
- Joint probability distribution
- Logic
- Market monetarism
- Mathematics
- Mid Term
- Octalysis Framework
- Omniment
- On the importance of Less Wrong, or another single conversational locus
- Our community should relocate to a country other than the US
- Paul Christiano's AI control blog
- People
- Philosophy
- Predictions For 2017
- Project proposal: Intro to the Universal Property
- Rationality
- The plan experiment
- Vitamin D
- Why waiting to donate harms charities

## Nate's ruminations

- Replacing Guilt
In my experience, many people are motivated primarily by either guilt, shame, or some combination of…

- Nate Soares- The Stamp Collector
Once upon a time, a group of naïve philosophers found a robot that collected trinkets. Well, more sp…

- Nate Soares - You're allowed to fight for something
The first sort of guilt I want to address is the listless guilt, that vague feeling one gets after p…

- Nate Soares

- The Stamp Collector

## Processes of Rationality

- Being Strategic: The Very Idea
Being strategic (here synonymous with "rationality" and "optimization") means systematically working…

- Nicolas Bourbaki - Noticing Preferences - Nicolas Bourbaki
- Trying Things
Optimization requires trying things and being wrong.

- Nicolas Bourbaki

## An Effective Altruist's Guide to the Future

- Resources and the future
Resource constraints are a widely held concern. Which are most likely to be limiting factors, and what can we do to relax those limits?

- Eric Bruylant- Civilization scale energy
What are the main options for powering civilization, and how do they compare?

- Eric Bruylant- Grid scale storage
Scalable energy storage is required if civilization's switches to primarily renewables in order to keep the grid powered at night. What are the options and how do they compare?

- Eric Bruylant

- Grid scale storage

- Civilization scale energy

## Improve comments by tagging claims

## 43m

- Arbital biographies
As a very strong default (presently an absolute rule), Joe Smith's page only says nice things about Joe. Even if a negative fact is true, it doesn't go on Joe's page.

- Eliezer Yudkowsky

## A beginner's guide to explaining things

- An algorithm for creating good explanations
Excellent explanations can break the rules, but it helps to know those rules in the first place. Following these steps will help you create explanations that are solid and that avoid common mistakes.

- Duncan Sabien - Core concepts for creating good explanations
A moderately deep dive into the concepts that underlie understanding and how to create it.

- Duncan Sabien - Examples of good explanations
A representative variety of good explanations, along with analysis of what each explanation does well (or not).

- Duncan Sabien - Tools and tactics for improving explanations
A list of some of the most useful and generally applicable techniques for creating understanding.

- Duncan Sabien

## AI alignment

- AI alignment open problem
Tag for open problems under AI alignment.

- Eliezer Yudkowsky - AI arms races
AI arms races are bad

- Eliezer Yudkowsky - Advanced safety
An agent is *really* safe when it has the capacity to do anything, but chooses to do what the programmer wants.

- Eliezer Yudkowsky- AI safety mindset
Asking how AI designs could go wrong, instead of imagining them going right.

- Eliezer Yudkowsky- Ad-hoc hack (alignment theory)
A "hack" is when you alter the behavior of your AI in a way that defies, or doesn't correspond to, a principled approach for that problem.

- Eliezer Yudkowsky - Directing, vs. limiting, vs. opposing
Getting the AI to compute the right action in a domain; versus getting the AI to not compute at all in an unsafe domain; versus trying to prevent the AI from acting successfully. (Prefer 1 & 2.)

- Eliezer Yudkowsky - Don't try to solve the entire alignment problem
New to AI alignment theory? Want to work in this area? Already been working in it for years? Don't try to solve the entire alignment problem with your next good idea!

- Eliezer Yudkowsky - Flag the load-bearing premises
If somebody says, "This AI safety plan is going to fail, because X" and you reply, "Oh, that's fine because of Y and Z", then you'd better clearly flag Y and Z as "load-bearing" parts of your plan.

- Eliezer Yudkowsky - Show me what you've broken
To demonstrate competence at computer security, or AI alignment, think in terms of breaking proposals and finding technically demonstrable flaws in them.

- Eliezer Yudkowsky - Valley of Dangerous Complacency
When the AGI works often enough that you let down your guard, but it still has bugs. Imagine a robotic car that almost always steers perfectly, but sometimes heads off a cliff.

- Eliezer Yudkowsky

- Ad-hoc hack (alignment theory)
- Actual effectiveness
If you want the AI's so-called 'utility function' to actually be steering the AI, you need to think about how it meshes up with beliefs, or what gets output to actions.

- Eliezer Yudkowsky - Context disaster
Some possible designs cause your AI to behave nicely while developing, and behave a lot less nicely when it's smarter.

- Eliezer Yudkowsky - Distinguish which advanced-agent properties lead to the foreseeable difficulty
Say what kind of AI, or threshold level of intelligence, or key type of advancement, first produces the difficulty or challenge you're talking about.

- Eliezer Yudkowsky - Goodhart's Curse
The Optimizer's Curse meets Goodhart's Law. For example, if our values are V, and an AI's utility function U is a proxy for V, optimizing for high U seeks out 'errors'--that is, high values of U - V.

- Eliezer Yudkowsky - Goodness estimate biaser
Some of the main problems in AI alignment can be seen as scenarios where actual goodness is likely to be systematically lower than a broken way of estimating goodness.

- Eliezer Yudkowsky - Methodology of foreseeable difficulties
Building a nice AI is likely to be hard enough, and contain enough gotchas that won't show up in the AI's early days, that we need to foresee problems coming in advance.

- Eliezer Yudkowsky - Methodology of unbounded analysis
What we do and don't understand how to do, using unlimited computing power, is a critical distinction and important frontier.

- Eliezer Yudkowsky- AIXI
How to build an (evil) superintelligent AI using unlimited computing power and one page of Python code.

- Eliezer Yudkowsky- AIXI-tl
A time-bounded version of the ideal agent AIXI that uses an impossibly large finite computer instead of a hypercomputer.

- Eliezer Yudkowsky

- AIXI-tl
- Cartesian agent
Agents separated from their environments by impermeable barriers through which only sensory information can enter and motor output can exit.

- Eliezer Yudkowsky- Cartesian agent-environment boundary
If your agent is separated from the environment by an absolute border that can only be crossed by sensory information and motor outputs, it might just be a Cartesian agent.

- Eliezer Yudkowsky

- Cartesian agent-environment boundary
- Hypercomputer
Some formalisms demand computers larger than the limit of all finite computers

- Eliezer Yudkowsky - Mechanical Turk (example)
The 19th-century chess-playing automaton known as the Mechanical Turk actually had a human operator inside. People at the time had interesting thoughts about the possibility of mechanical chess.

- Eliezer Yudkowsky - No-Free-Lunch theorems are often irrelevant
There's often a theorem proving that some problem has no optimal answer across every possible world. But this may not matter, since the real world is a special case. (E.g., a low-entropy universe.)

- Eliezer Yudkowsky - Solomonoff induction
A simple way to superintelligently predict sequences of data, given unlimited computing power.

- Eliezer Yudkowsky- Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math

- Eliezer Yudkowsky

- Solomonoff induction: Intro Dialogue (Math 2)
- Unphysically large finite computer
The imaginary box required to run programs that require impossibly large, but finite, amounts of computing power.

- Eliezer Yudkowsky

- AIXI
- Nearest unblocked strategy
If you patch an agent's preference framework to avoid an undesirable solution, what can you expect to happen?

- Eliezer Yudkowsky - Optimization daemons
When you optimize something so hard that it crystalizes into an optimizer, like the way natural selection optimized apes so hard they turned into human-level intelligences

- Eliezer Yudkowsky - Safe but useless
Sometimes, at the end of locking down your AI so that it seems extremely safe, you'll end up with an AI that can't be used to do anything interesting.

- Eliezer Yudkowsky

- AI safety mindset
- Complexity of value
There's no simple way to describe the goals we want Artificial Intelligences to want.

- Eliezer Yudkowsky- Meta-rules for (narrow) value learning are still unsolved
We don't currently know a simple meta-utility function that would take in observation of humans and spit out our true values, or even a good target for a Task AGI.

- Eliezer Yudkowsky - Underestimating complexity of value because goodness feels like a simple property
When you just want to yell at the AI, "Just do normal high-value X, dammit, not weird low-value X!" and that 'high versus low value' boundary is way more complicated than your brain wants to think.

- Eliezer Yudkowsky

- Meta-rules for (narrow) value learning are still unsolved
- Coordinative AI development hypothetical
What would safe AI development look like if we didn't have to worry about anything else?

- Eliezer Yudkowsky - Correlated coverage
In which parts of AI alignment can we hope that getting many things right, will mean the AI gets everything right?

- Eliezer Yudkowsky - Corrigibility
"I can't let you do that, Dave."

- Nate Soares- Averting instrumental pressures
Almost-any utility function for an AI, whether the target is diamonds or paperclips or eudaimonia, implies subgoals like rapidly self-improving and refusing to shut down. Can we make that not happen?

- Eliezer Yudkowsky - Averting the convergent instrumental strategy of self-improvement
We probably want the first AGI to *not* improve as fast as possible, but improving as fast as possible is a convergent strategy for accomplishing most things.

- Eliezer Yudkowsky - Hard problem of corrigibility
Can you build an agent that reasons as if it knows itself to be incomplete and sympathizes with your wanting to rebuild or correct it?

- Eliezer Yudkowsky - Interruptibility
A subproblem of corrigibility under the machine learning paradigm: when the agent is interrupted, it must not learn to prevent future interruptions.

- Eliezer Yudkowsky - Problem of fully updated deference
Why moral uncertainty doesn't stop an AI from defending its off-switch.

- Eliezer Yudkowsky - Programmer deception
Programmer deception is when the AI's decision process leads it to optimize for an instrumental goal…

- Eliezer Yudkowsky- Cognitive steganography
Disaligned AIs that are modeling human psychology and trying to deceive their programmers will want to hide their internal thought processes from their programmers.

- Eliezer Yudkowsky

- Cognitive steganography
- Shutdown problem
How to build an AGI that lets you shut it down, despite the obvious fact that this will interfere with whatever the AGI's goals are.

- Eliezer Yudkowsky- You can't get the coffee if you're dead
An AI given the goal of 'get the coffee' can't achieve that goal if it has been turned off; so even an AI whose goal is just to fetch the coffee may try to avert a shutdown button being pressed.

- Eliezer Yudkowsky

- You can't get the coffee if you're dead
- User manipulation
If not otherwise averted, many of an AGI's desired outcomes are likely to interact with users and hence imply an incentive to manipulate users.

- Eliezer Yudkowsky- User maximization
A sub-principle of avoiding user manipulation - if you see an argmax over X or 'optimize X' instruction and X includes a user interaction, you've just told the AI to optimize the user.

- Eliezer Yudkowsky

- User maximization
- Utility indifference
How can we make an AI indifferent to whether we press a button that changes its goals?

- Eliezer Yudkowsky

- Averting instrumental pressures
- Development phase unpredictable
Several proposed problems in advanced safety are alleged to be difficult because they depend on some…

- Eliezer Yudkowsky- Unforeseen maximum
When you tell AI to produce world peace and it kills everyone. (Okay, some SF writers saw that one coming.)

- Eliezer Yudkowsky- Missing the weird alternative
People might systematically overlook "make tiny molecular smileyfaces" as a way of "producing smiles", because our brains automatically search for high-utility-to-us ways of "producing smiles".

- Eliezer Yudkowsky

- Missing the weird alternative

- Unforeseen maximum
- Difficulty of AI alignment
How hard is it exactly to point an Artificial General Intelligence in an intuitively okay direction?

- Eliezer Yudkowsky - Executable philosophy
Philosophical discourse aimed at producing a trustworthy answer or meta-answer, in limited time, which can used in constructing an Artificial Intelligence.

- Eliezer Yudkowsky - Glossary (Value Alignment Theory)
Words that have a special meaning in the context of creating nice AIs.

- Eliezer Yudkowsky- 'Concept'
In the context of Artificial Intelligence, a 'concept' is a category, something that identifies thingies as being inside or outside the concept.

- Eliezer Yudkowsky - Cognitive domain
An allegedly compact unit of knowledge, such that ideas inside the unit interact mainly with each other and less with ideas in other domains.

- Eliezer Yudkowsky- Distances between cognitive domains
Often in AI alignment we want to ask, "How close is 'being able to do X' to 'being able to do Y'?"

- Eliezer Yudkowsky

- Distances between cognitive domains
- Friendly AI
Old terminology for an AI whose preferences have been successfully aligned with idealized human values.

- Eliezer Yudkowsky

- 'Concept'
- Identifying ambiguous inductions
What do a "red strawberry", a "red apple", and a "red cherry" have in common that a "yellow carrot" doesn't? Are they "red fruits" or "red objects"?

- Eliezer Yudkowsky - Informed oversight
Incentivize a reinforcement learner that's less smart than you to accomplish some task

- Jessica Taylor - Intended goal
Definition. An "intended goal" refers to the intuitive intention in the mind of a human programmer …

- Eliezer Yudkowsky - Linguistic conventions in value alignment
How and why to use precise language and words with special meaning when talking about value alignment.

- Eliezer Yudkowsky- Utility
What is "utility" in the context of Value Alignment Theory?

- Eliezer Yudkowsky

- Utility
- List: value-alignment subjects
Bullet point list of core VAT subjects.

- Eliezer Yudkowsky - Mindcrime
Might a machine intelligence contain vast numbers of unhappy conscious subprocesses?

- Eliezer Yudkowsky- Mindcrime: Introduction
The more predictive accuracy we want from a model, the more detailed the model becomes. A very roug…

- Eliezer Yudkowsky - Nonperson predicate
If we knew which computations were definitely not people, we could tell AIs which programs they were definitely allowed to compute.

- Eliezer Yudkowsky

- Mindcrime: Introduction
- Modeling distant superintelligences
The several large problems that might occur if an AI starts to think about alien superintelligences.

- Eliezer Yudkowsky- Distant superintelligences can coerce the most probable environment of your AI
Distant superintelligences may be able to hack your local AI, if your AI's preference framework depends on its most probable environment.

- Eliezer Yudkowsky

- Distant superintelligences can coerce the most probable environment of your AI
- Natural language understanding of "right" will yield normativity
What will happen if you tell an advanced agent to do the "right" thing?

- Eliezer Yudkowsky - Nick Bostrom's book Superintelligence
The current best book-form introduction to AI alignment theory.

- Eliezer Yudkowsky - Object-level vs. indirect goals
Difference between "give Alice the apple" and "give Alice what she wants".

- Eliezer Yudkowsky - Patch resistance
One does not simply solve the value alignment problem.

- Eliezer Yudkowsky- Unforeseen maximum
When you tell AI to produce world peace and it kills everyone. (Okay, some SF writers saw that one coming.)

- Eliezer Yudkowsky- Missing the weird alternative
People might systematically overlook "make tiny molecular smileyfaces" as a way of "producing smiles", because our brains automatically search for high-utility-to-us ways of "producing smiles".

- Eliezer Yudkowsky

- Missing the weird alternative

- Unforeseen maximum
- Principles in AI alignment
A 'principle' of AI alignment is a very general design goal like 'understand what the heck is going on inside the AI' that has informed a wide set of specific design proposals.

- Eliezer Yudkowsky- Minimality principle
The first AGI ever built should save the world in a way that requires the least amount of the least dangerous cognition.

- Eliezer Yudkowsky - Non-adversarial principle
At no point in constructing an Artificial General Intelligence should we construct a computation that tries to hurt us, and then try to stop it from hurting us.

- Eliezer Yudkowsky- Directing, vs. limiting, vs. opposing
Getting the AI to compute the right action in a domain; versus getting the AI to not compute at all in an unsafe domain; versus trying to prevent the AI from acting successfully. (Prefer 1 & 2.)

- Eliezer Yudkowsky - Generalized principle of cognitive alignment
When we're asking how we want the AI to think about an alignment problem, one source of inspiration is trying to have the AI mirror our own thoughts about that problem.

- Eliezer Yudkowsky - Niceness is the first line of defense
The *first* line of defense in dealing with any partially superhuman AI system advanced enough to possibly be dangerous is that it does not *want* to hurt you or defeat your safety measures.

- Eliezer Yudkowsky - Omnipotence test for AI safety
Would your AI produce disastrous outcomes if it suddenly gained omnipotence and omniscience? If so, why did you program something that *wants* to hurt you and is held back only by lacking the power?

- Eliezer Yudkowsky - The AI must tolerate your safety measures
A corollary of the nonadversarial principle is that "The AI must tolerate your safety measures."

- Eliezer Yudkowsky

- Directing, vs. limiting, vs. opposing
- Separation from hyperexistential risk
The AI should be widely separated in the design space from any AI that would constitute a "hyperexistential risk" (anything worse than death).

- Eliezer Yudkowsky - Understandability principle
The more you understand what the heck is going on inside your AI, the safer you are.

- Eliezer Yudkowsky- Effability principle
You are safer the more you understand the inner structure of how your AI thinks; the better you can describe the relation of smaller pieces of the AI's thought process.

- Eliezer Yudkowsky

- Effability principle

- Minimality principle
- Programmer
Who is building these advanced agents?

- Eliezer Yudkowsky - Relevant limited AI
Can we have a limited AI, that's nonetheless relevant?

- Eliezer Yudkowsky - Relevant powerful agent
An agent is relevant if it completely changes the course of history.

- Eliezer Yudkowsky - Relevant powerful agents will be highly optimized
The probability that an agent that is cognitively powerful enough to be relevant to existential outc…

- Eliezer Yudkowsky - Reliable prediction
How can we train predictors that reliably predict observable phenomena such as human behavior?

- Jessica Taylor - Researchers in value alignment theory
Who's working full-time in value alignment theory?

- Eliezer Yudkowsky- Nick Bostrom
Nick Bostrom, secretly the inventor of Friendly AI

- Eliezer Yudkowsky

- Nick Bostrom
- Safe impact measure
What can we measure to make sure an agent is acting in a safe manner?

- Eliezer Yudkowsky - Safe training procedures for human-imitators
How does one train a reinforcement learner to act like a human?

- Jessica Taylor - Selective similarity metrics for imitation
Can we make human-imitators more efficient by scoring them more heavily on imitating the aspects of human behavior we care about more?

- Jessica Taylor - Some computations are people
It's possible to have a conscious person being simulated inside a computer or other substrate.

- Eliezer Yudkowsky - Strategic AGI typology
What broad types of advanced AIs, corresponding to which strategic scenarios, might it be possible or wise to create?

- Eliezer Yudkowsky- Autonomous AGI
The hardest possible class of Friendly AI to build, with the least moral hazard; an AI intended to neither require nor accept further direction.

- Eliezer Yudkowsky - Known-algorithm non-self-improving agent
Possible advanced AIs that aren't self-modifying, aren't self-improving, and where we know and understand all the component algorithms.

- Eliezer Yudkowsky - Oracle
System designed to safely answer questions.

- Eliezer Yudkowsky- Zermelo-Fraenkel provability oracle
We might be able to build a system that can safely inform us that a theorem has a proof in set theory, but we can't see how to use that capability to save the world.

- Eliezer Yudkowsky

- Zermelo-Fraenkel provability oracle
- Task-directed AGI
An advanced AI that's meant to pursue a series of limited-scope goals given it by the user. In Bostrom's terminology, a Genie.

- Eliezer Yudkowsky- Behaviorist genie
An advanced agent that's forbidden to model minds in too much detail.

- Eliezer Yudkowsky - Boxed AI
Idea: what if we limit how AI can interact with the world. That'll make it safe, right??

- Eliezer Yudkowsky- Zermelo-Fraenkel provability oracle
We might be able to build a system that can safely inform us that a theorem has a proof in set theory, but we can't see how to use that capability to save the world.

- Eliezer Yudkowsky

- Zermelo-Fraenkel provability oracle
- Conservative concept boundary
Given N example burritos, draw a boundary around what is a 'burrito' that is relatively simple and allows as few positive instances as possible. Helps make sure the next thing generated is a burrito.

- Eliezer Yudkowsky - Epistemic exclusion
How would you build an AI that, no matter what else it learned about the world, never knew or wanted to know what was inside your basement?

- Eliezer Yudkowsky - Faithful simulation
How would you identify, to a Task AGI (aka Genie), the problem of scanning a human brain, and then running a sufficiently accurate simulation of it for the simulation to not be crazy or psychotic?

- Eliezer Yudkowsky - Limited AGI
Task-based AGIs don't need unlimited cognitive and material powers to carry out their Tasks; which means their powers can potentially be limited.

- Eliezer Yudkowsky - Low impact
The open problem of having an AI carry out tasks in ways that cause minimum side effects and change as little of the rest of the universe as possible.

- Eliezer Yudkowsky- Abortable plans
Plans that can be undone, or switched to having low further impact. If the AI builds abortable nanomachines, they'll have a quiet self-destruct option that includes any replicated nanomachines.

- Eliezer Yudkowsky - Shutdown utility function
A special case of a low-impact utility function where you just want the AGI to switch itself off harmlessly (and not create subagents to make absolutely sure it stays off, etcetera).

- Eliezer Yudkowsky

- Abortable plans
- Mild optimization
An AGI which, if you ask it to paint one car pink, just paints one car pink and doesn't tile the universe with pink-painted cars, because it's not trying *that* hard to max out its car-painting score.

- Eliezer Yudkowsky - Open subproblems in aligning a Task-based AGI
Open research problems, especially ones we can model today, in building an AGI that can "paint all cars pink" without turning its future light cone into pink-painted cars.

- Eliezer Yudkowsky - Oracle
System designed to safely answer questions.

- Eliezer Yudkowsky- Zermelo-Fraenkel provability oracle
We might be able to build a system that can safely inform us that a theorem has a proof in set theory, but we can't see how to use that capability to save the world.

- Eliezer Yudkowsky

- Zermelo-Fraenkel provability oracle
- Querying the AGI user
Postulating that an advanced agent will check something with its user, probably comes with some standard issues and gotchas (e.g., prioritizing what to query, not manipulating the user, etc etc).

- Eliezer Yudkowsky - Safe plan identification and verification
On a particular task or problem, the issue of how to communicate to the AGI what you want it to do and all the things you don't want it to do.

- Eliezer Yudkowsky- Do-What-I-Mean hierarchy
Successive levels of "Do What I Mean" or AGIs that understand their users increasingly well

- Eliezer Yudkowsky

- Do-What-I-Mean hierarchy
- Task (AI goal)
When building the first AGIs, it may be wiser to assign them only goals that are bounded in space and time, and can be satisfied by bounded efforts.

- Eliezer Yudkowsky - Task identification problem
If you have a task-based AGI (Genie) then how do you pinpoint exactly what you want it to do (and not do)?

- Eliezer Yudkowsky- Look where I'm pointing, not at my finger
When trying to communicate the concept "glove", getting the AGI to focus on "gloves" rather than "my user's decision to label something a glove" or "anything that depresses the glove-labeling button".

- Eliezer Yudkowsky

- Look where I'm pointing, not at my finger

- Behaviorist genie

- Autonomous AGI
- Strong cognitive uncontainability
An advanced agent can win in ways humans can't understand in advance.

- Eliezer Yudkowsky - Sufficiently optimized agents appear coherent
If you could think as well as a superintelligence, you'd be at least that smart yourself.

- Eliezer Yudkowsky - Task-directed AGI
An advanced AI that's meant to pursue a series of limited-scope goals given it by the user. In Bostrom's terminology, a Genie.

- Eliezer Yudkowsky- Behaviorist genie
An advanced agent that's forbidden to model minds in too much detail.

- Eliezer Yudkowsky - Boxed AI
Idea: what if we limit how AI can interact with the world. That'll make it safe, right??

- Eliezer Yudkowsky- Zermelo-Fraenkel provability oracle

- Zermelo-Fraenkel provability oracle
- Conservative concept boundary
Given N example burritos, draw a boundary around what is a 'burrito' that is relatively simple and allows as few positive instances as possible. Helps make sure the next thing generated is a burrito.

- Eliezer Yudkowsky - Epistemic exclusion
How would you build an AI that, no matter what else it learned about the world, never knew or wanted to know what was inside your basement?

- Eliezer Yudkowsky - Faithful simulation
How would you identify, to a Task AGI (aka Genie), the problem of scanning a human brain, and then running a sufficiently accurate simulation of it for the simulation to not be crazy or psychotic?

- Eliezer Yudkowsky - Limited AGI
Task-based AGIs don't need unlimited cognitive and material powers to carry out their Tasks; which means their powers can potentially be limited.

- Eliezer Yudkowsky - Low impact
The open problem of having an AI carry out tasks in ways that cause minimum side effects and change as little of the rest of the universe as possible.

- Eliezer Yudkowsky- Abortable plans
Plans that can be undone, or switched to having low further impact. If the AI builds abortable nanomachines, they'll have a quiet self-destruct option that includes any replicated nanomachines.

- Eliezer Yudkowsky - Shutdown utility function
A special case of a low-impact utility function where you just want the AGI to switch itself off harmlessly (and not create subagents to make absolutely sure it stays off, etcetera).

- Eliezer Yudkowsky

- Abortable plans
- Mild optimization
An AGI which, if you ask it to paint one car pink, just paints one car pink and doesn't tile the universe with pink-painted cars, because it's not trying *that* hard to max out its car-painting score.

- Eliezer Yudkowsky - Open subproblems in aligning a Task-based AGI
Open research problems, especially ones we can model today, in building an AGI that can "paint all cars pink" without turning its future light cone into pink-painted cars.

- Eliezer Yudkowsky - Oracle
System designed to safely answer questions.

- Eliezer Yudkowsky- Zermelo-Fraenkel provability oracle

- Zermelo-Fraenkel provability oracle
- Querying the AGI user
Postulating that an advanced agent will check something with its user, probably comes with some standard issues and gotchas (e.g., prioritizing what to query, not manipulating the user, etc etc).

- Eliezer Yudkowsky - Safe plan identification and verification
On a particular task or problem, the issue of how to communicate to the AGI what you want it to do and all the things you don't want it to do.

- Eliezer Yudkowsky- Do-What-I-Mean hierarchy
Successive levels of "Do What I Mean" or AGIs that understand their users increasingly well

- Eliezer Yudkowsky

- Do-What-I-Mean hierarchy
- Task (AI goal)
When building the first AGIs, it may be wiser to assign them only goals that are bounded in space and time, and can be satisfied by bounded efforts.

- Eliezer Yudkowsky - Task identification problem
If you have a task-based AGI (Genie) then how do you pinpoint exactly what you want it to do (and not do)?

- Eliezer Yudkowsky- Look where I'm pointing, not at my finger
When trying to communicate the concept "glove", getting the AGI to focus on "gloves" rather than "my user's decision to label something a glove" or "anything that depresses the glove-labeling button".

- Eliezer Yudkowsky

- Look where I'm pointing, not at my finger

- Behaviorist genie
- The rocket alignment problem
If people talked about the problem of space travel the way they talked about AI...

- Eliezer Yudkowsky - Theory of (advanced) agents
One of the research subproblems of building powerful nice AIs, is the theory of (sufficiently advanced) minds in general.

- Eliezer Yudkowsky- Advanced agent properties
How smart does a machine intelligence need to be, for its niceness to become an issue? "Advanced" is a broad term to cover cognitive abilities such that we'd need to start considering AI alignment.

- Eliezer Yudkowsky- Advanced nonagent
Hypothetically, cognitively powerful programs that don't follow the loop of "observe, learn, model the consequences, act, observe results" that a standard "agent" would.

- Eliezer Yudkowsky - Artificial General Intelligence
An AI which has the same kind of "significantly more general" intelligence that humans have compared to chimpanzees; it can learn new domains, like we can.

- Eliezer Yudkowsky - Big-picture strategic awareness
We start encountering new AI alignment issues at the point where a machine intelligence recognizes the existence of a real world, the existence of programmers, and how these relate to its goals.

- Eliezer Yudkowsky - Cognitive uncontainability
'Cognitive uncontainability' is when we can't hold all of an agent's possibilities inside our own minds.

- Eliezer Yudkowsky- Rich domain
A domain is 'rich', relative to our own intelligence, to the extent that (1) its [ search space] is …

- Eliezer Yudkowsky- Almost all real-world domains are rich
Anything you're trying to accomplish in the real world can potentially be accomplished in a *lot* of different ways.

- Eliezer Yudkowsky - Logical game
Game's mathematical structure at its purest form.

- Eliezer Yudkowsky

- Almost all real-world domains are rich

- Rich domain
- Consequentialist cognition
The cognitive ability to foresee the consequences of actions, prefer some outcomes to others, and output actions leading to the preferred outcomes.

- Eliezer Yudkowsky - Corporations vs. superintelligences
Corporations have relatively few of the advanced-agent properties that would allow one mistake in aligning a corporation to immediately kill all humans and turn the future light cone into paperclips.

- Eliezer Yudkowsky - Epistemic and instrumental efficiency
An efficient agent never makes a mistake you can predict. You can never successfully predict a directional bias in its estimates.

- Eliezer Yudkowsky- Time-machine metaphor for efficient agents
Don't imagine a paperclip maximizer as a mind. Imagine it as a time machine that always spits out the output leading to the greatest number of future paperclips.

- Eliezer Yudkowsky

- Time-machine metaphor for efficient agents
- General intelligence
Compared to chimpanzees, humans seem to be able to learn a much wider variety of domains. We have 'significantly more generally applicable' cognitive abilities, aka 'more general intelligence'.

- Eliezer Yudkowsky - Infrahuman, par-human, superhuman, efficient, optimal
A categorization of AI ability levels relative to human, with some gotchas in the ordering. E.g., in simple domains where humans can play optimally, optimal play is not superhuman.

- Eliezer Yudkowsky - Intelligence explosion
What happens if a self-improving AI gets to the point where each amount x of self-improvement triggers >x further self-improvement, and it stays that way for a while.

- Eliezer Yudkowsky - Real-world domain
Some AIs play chess, some AIs play Go, some AIs drive cars. These different 'domains' present different options. All of reality, in all its messy entanglement, is the 'real-world domain'.

- Eliezer Yudkowsky - Standard agent properties
What's a Standard Agent, and what can it do?

- Eliezer Yudkowsky- Bounded agent
An agent that operates in the real world, using realistic amounts of computing power, that is uncertain of its environment, etcetera.

- Eliezer Yudkowsky

- Bounded agent
- Sufficiently advanced Artificial Intelligence
'Sufficiently advanced Artificial Intelligences' are AIs with enough 'advanced agent properties' that we start needing to do 'AI alignment' to them.

- Eliezer Yudkowsky - Superintelligent
A "superintelligence" is strongly superhuman (strictly higher-performing than any and all humans) on every cognitive problem.

- Eliezer Yudkowsky - Vingean uncertainty
You can't predict the exact actions of an agent smarter than you - so is there anything you _can_ say about them?

- Eliezer Yudkowsky- Deep Blue
The chess-playing program, built by IBM, that first won the world chess championship from Garry Kasparov in 1996.

- Eliezer Yudkowsky - Vinge's Law
You can't predict exactly what someone smarter than you would do, because if you could, you'd be that smart yourself.

- Eliezer Yudkowsky

- Deep Blue

- Advanced nonagent
- Instrumental convergence
Some strategies can help achieve most possible simple goals. E.g., acquiring more computing power or more material resources. By default, unless averted, we can expect advanced AIs to do that.

- Eliezer Yudkowsky- Convergent instrumental strategies
Paperclip maximizers can make more paperclips by improving their cognitive abilities or controlling more resources. What other strategies would almost-any AI try to use?

- Eliezer Yudkowsky- Consequentialist preferences are reflectively stable by default
Gandhi wouldn't take a pill that made him want to kill people, because he knows in that case more people will be murdered. A paperclip maximizer doesn't want to stop maximizing paperclips.

- Eliezer Yudkowsky - Convergent strategies of self-modification
The strategies we'd expect to be employed by an AI that understands the relevance of its code and hardware to achieving its goals, which therefore has subgoals about its code and hardware.

- Eliezer Yudkowsky

- Consequentialist preferences are reflectively stable by default
- Instrumental
What is "instrumental" in the context of Value Alignment Theory?

- Eliezer Yudkowsky - Instrumental pressure
A consequentialist agent will want to bring about certain instrumental events that will help to fulfill its goals.

- Eliezer Yudkowsky - Paperclip maximizer
This agent will not stop until the entire universe is filled with paperclips.

- Eliezer Yudkowsky- Paperclip
A configuration of matter that we'd see as being worthless even from a very cosmopolitan perspective.

- Eliezer Yudkowsky - Random utility function
A 'random' utility function is one chosen at random according to some simple probability measure (e.g. weight by Kolmorogov complexity) on a logical space of formal utility functions.

- Eliezer Yudkowsky

- Paperclip
- You can't get more paperclips that way
Most arguments that "A paperclip maximizer could get more paperclips by (doing nice things)" are flawed.

- Eliezer Yudkowsky

- Convergent instrumental strategies
- Orthogonality Thesis
Will smart AIs automatically become benevolent, or automatically become hostile? Or do different AI designs imply different goals?

- Eliezer Yudkowsky- Instrumental goals are almost-equally as tractable as terminal goals
Getting the milk from the refrigerator because you want to drink it, is not vastly harder than getting the milk from the refrigerator because you inherently desire it.

- Eliezer Yudkowsky - Mind design space is wide
Imagine all human beings as one tiny dot inside a much vaster sphere of possibilities for "The space of minds in general." It is wiser to make claims about *some* minds than *all* minds.

- Eliezer Yudkowsky - Paperclip maximizer
This agent will not stop until the entire universe is filled with paperclips.

- Eliezer Yudkowsky- Paperclip
A configuration of matter that we'd see as being worthless even from a very cosmopolitan perspective.

- Eliezer Yudkowsky - Random utility function
A 'random' utility function is one chosen at random according to some simple probability measure (e.g. weight by Kolmorogov complexity) on a logical space of formal utility functions.

- Eliezer Yudkowsky

- Paperclip

- Instrumental goals are almost-equally as tractable as terminal goals

- Advanced agent properties
- Unforeseen maximum
When you tell AI to produce world peace and it kills everyone. (Okay, some SF writers saw that one coming.)

- Eliezer Yudkowsky- Missing the weird alternative
People might systematically overlook "make tiny molecular smileyfaces" as a way of "producing smiles", because our brains automatically search for high-utility-to-us ways of "producing smiles".

- Eliezer Yudkowsky

- Missing the weird alternative
- VAT playpen
Playpen page for VAT domain.

- Alexei Andreev - Value
The word 'value' in the phrase 'value alignment' is a metasyntactic variable that indicates the speaker's future goals for intelligent life.

- Eliezer Yudkowsky- 'Beneficial'
Really actually good. A metasyntactic variable to mean "favoring whatever the speaker wants ideally to accomplish", although different speakers have different morals and metaethics.

- Eliezer Yudkowsky - 'Detrimental'
The opposite of beneficial.

- Eliezer Yudkowsky - Coherent extrapolated volition (alignment target)
A proposed direction for an extremely well-aligned autonomous superintelligence - do what humans would want, if we knew what the AI knew, thought that fast, and understood ourselves.

- Eliezer Yudkowsky - Cosmopolitan value
Intuitively: Value as seen from a broad, embracing standpoint that is aware of how other entities may not always be like us or easily understandable to us, yet still worthwhile.

- Eliezer Yudkowsky - Extrapolated volition (normative moral theory)
If someone asks you for orange juice, and you know that the refrigerator contains no orange juice, should you bring them lemonade?

- Eliezer Yudkowsky- Rescuing the utility function
If your utility function values 'heat', and then you discover to your horror that there's no ontologically basic heat, switch to valuing disordered kinetic energy. Likewise 'free will' or 'people'.

- Eliezer Yudkowsky

- Rescuing the utility function
- Immediate goods
One of the potential views on 'value' in the value alignment problem is that what we should want fro…

- Eliezer Yudkowsky - William Frankena's list of terminal values
Life, consciousness, and activity; health and strength; pleasures and satisfactions of all or certain kinds; happiness, beatitude, contentment, etc.; truth; knowledge and true opinions...

- Eliezer Yudkowsky

- 'Beneficial'
- Value achievement dilemma
How can Earth-originating intelligent life achieve most of its potential value, whether by AI or otherwise?

- Eliezer Yudkowsky- Aligning an AGI adds significant development time
Aligning an advanced AI foreseeably involves extra code and extra testing and not being able to do everything the fastest way, so it takes longer.

- Eliezer Yudkowsky - Coordinative AI development hypothetical
What would safe AI development look like if we didn't have to worry about anything else?

- Eliezer Yudkowsky - Cosmic endowment
The 'cosmic endowment' consists of all the stars that could be reached from probes originating on Earth; the sum of all matter and energy potentially available to be transformed into life and fun.

- Eliezer Yudkowsky - Moral hazards in AGI development
"Moral hazard" is when owners of an advanced AGI give in to the temptation to do things with it that the rest of us would regard as 'bad', like, say, declaring themselves God-Emperor.

- Eliezer Yudkowsky - Pivotal event
Which types of AIs, if they work, can do things that drastically change the nature of the further game?

- Eliezer Yudkowsky

- Aligning an AGI adds significant development time
- Value alignment problem
You want to build an advanced AI with the right values... but how?

- Eliezer Yudkowsky- Preference framework
What's the thing an agent uses to compare its preferences?

- Eliezer Yudkowsky- Attainable optimum
The 'attainable optimum' of an agent's preferences is the best that agent can actually do given its finite intelligence and resources (as opposed to the global maximum of those preferences).

- Eliezer Yudkowsky - Meta-utility function
Preference frameworks built out of simple utility functions, but where, e.g., the 'correct' utility function for a possible world depends on whether a button is pressed.

- Eliezer Yudkowsky - Moral uncertainty
A meta-utility function in which the utility function as usually considered, takes on different values in different possible worlds, potentially distinguishable by evidence.

- Eliezer Yudkowsky- Ideal target
The 'ideal target' of a meta-utility function is the value the ground-level utility function would take on if the agent updated on all possible evidence; the 'true' utilities under moral uncertainty.

- Eliezer Yudkowsky

- Ideal target

- Attainable optimum
- Total alignment
We say that an advanced AI is "totally aligned" when it knows *exactly* which outcomes and plans are beneficial, with no further user input.

- Eliezer Yudkowsky

- Preference framework
- Value identification problem
The subproblem category of value alignment which deals with pinpointing valuable outcomes to an adva…

- Eliezer Yudkowsky- Edge instantiation
When you ask the AI to make people happy, and it tiles the universe with the smallest objects that can be happy.

- Eliezer Yudkowsky - Environmental goals
The problem of having an AI want outcomes that are out in the world, not just want direct sense events.

- Eliezer Yudkowsky - Goal-concept identification
Figuring out how to say "strawberry" to an AI that you want to bring you strawberries (and not fake plastic strawberries, either).

- Eliezer Yudkowsky - Happiness maximizer
It is sometimes proposed that we build an AI intended to maximize human happiness. (One early propo…

- Eliezer Yudkowsky - Identifying causal goal concepts from sensory data
If the intended goal is "cure cancer" and you show the AI healthy patients, it sees, say, a pattern of pixels on a webcam. How do you get to a goal concept *about* the real patients?

- Eliezer Yudkowsky - Ontology identification problem
How do we link an agent's utility function to its model of the world, when we don't know what that model will look like?

- Eliezer Yudkowsky- Diamond maximizer
How would you build an agent that made as much diamond material as possible, given vast computing power but an otherwise rich and complicated environment?

- Eliezer Yudkowsky - Ontology identification problem: Technical tutorial
Technical tutorial for ontology identification problem.

- Eliezer Yudkowsky

- Diamond maximizer

- Edge instantiation
- Vingean reflection
The problem of thinking about your future self when it's smarter than you.

- Eliezer Yudkowsky- Reflective consistency
A decision system is reflectively consistent if it can approve of itself, or approve the construction of similar decision systems (as well as perhaps approving other decision systems too).

- Eliezer Yudkowsky - Reflective stability
Wanting to think the way you currently think, building other agents and self-modifications that think the same way.

- Eliezer Yudkowsky- Consequentialist preferences are reflectively stable by default
Gandhi wouldn't take a pill that made him want to kill people, because he knows in that case more people will be murdered. A paperclip maximizer doesn't want to stop maximizing paperclips.

- Eliezer Yudkowsky - Other-izing (wanted: new optimization idiom)
Maximization isn't possible for bounded agents, and satisficing doesn't seem like enough. What other kind of 'izing' might be good for realistic, bounded agents?

- Eliezer Yudkowsky - Reflectively consistent degree of freedom
When an instrumentally efficient, self-modifying AI can be like X or like X' in such a way that X wants to be X and X' wants to be X', that's a reflectively consistent degree of freedom.

- Eliezer Yudkowsky- Humean degree of freedom
A concept includes 'Humean degrees of freedom' when the intuitive borders of the human version of that concept depend on our values, making that concept less natural for AIs to learn.

- Eliezer Yudkowsky - Value-laden
Cure cancer, but avoid any bad side effects? Categorizing "bad side effects" requires knowing what's "bad". If an agent needs to load complex human goals to evaluate something, it's "value-laden".

- Eliezer Yudkowsky

- Humean degree of freedom

- Consequentialist preferences are reflectively stable by default
- Tiling agents theory
The theory of self-modifying agents that build successors that are very similar to themselves, like repeating tiles on a tesselated plane.

- Eliezer Yudkowsky - Vinge's Principle
An agent building another agent must usually approve its design without knowing the agent's exact policy choices.

- Eliezer Yudkowsky

- Reflective consistency

## Accelerator Project

- A whirlwind tour
A rapid tour of Eric's thoughts on the accelerator project.

- Eric Bruylant - Funding governance filtering
Current-best guess at how these parts of organization will happen. ### Funding Eric provided initi…

- Eric Bruylant - The missing step between Zero and Hero
Creating a space for high potential people grow and improve the world at scale.

- Eric Bruylant - Who needs civilization?
The most common difference between my model and that of people I've discussed this with is that many…

- Eric Bruylant

## Active learning for opaque, powerful predictors

- Research directions in AI control
What research would best advance our understanding of AI control? I’ve been thinking about this qu…

- Paul Christiano

## Alexei Andreev

- Alexei Andreev: Personal
Fun things Alexei Andreev likes to do.

- Alexei Andreev

## Alexei's Blog

- Unemployment and Automation
#WORK IN PROGRESS I'm going to research and explore the topic of unemployment and automation. Quest…

- Alexei Andreev- Will Narrow AI Seriously Affect Long-Term Employment?
###Definitions: **Narrow AI** - artificial intelligence capable of doing a narrow task at a level …

- Alexei Andreev

- Will Narrow AI Seriously Affect Long-Term Employment?
- Working out, weightlifting, and nutrition
Just a collection of links / resources I'll be using to seed some content. # Working out ### How t…

- Alexei Andreev

## Algebraic field

- Field homomorphism is trivial or injective
Field homomorphisms preserve a *lot* of structure; they preserve so much structure that they are always either injective or totally boring.

- Patrick Stevens

## Ambitious vs. narrow value learning

- Imitation and justification
Suppose that I am training an AI system to play Go. One approach is to have the AI observe human m…

- Paul Christiano

## Anton Geraschenko

- Alternate
Clickbait?

- Anton Geraschenko

## Arbital

- Arbital Blog
Stay up to date on all things Arbital

- Alexei Andreev- Arbital: Solving online explanations
An explanation of Arbital's mid-term goals

- Eliezer Yudkowsky - Easier, better, faster, brighter
In which Arbital gets a makeover, shortcuts, quick feedback, and more!

- Alexei Andreev - Now I am become Life, the protector of worlds
In which Arbital is brought forth into the world.

- Alexei Andreev - Some men just want to watch the world learn
In which Arbital gets paths.

- Alexei Andreev - To math explanations and beyond!
In which Arbital doubles down on math explanations.

- Alexei Andreev

- Arbital: Solving online explanations
- Arbital Slack
Where the cool kids hang out.

- Eric Bruylant - Arbital archive
An archive of all public content on Arbital!

- Eric Bruylant - Arbital community input
Do you have ideas about how to improve Arbital which you think the community should discuss?

- Alexei Andreev- Children in a sidebar
I always forget to scroll all the way down. I shouldn't have to. Putting children in a sidebar allow…

- Olivia Schaefer - Page's title should always be capitalized
Vote "agree" if you think Arbital should enforce the first letter of a page title to always be capit…

- Alexei Andreev

- Children in a sidebar
- Arbital content license
What license does Arbital use for its content?

- Alexei Andreev - Arbital playpen
Want to test a feature? Feel free to edit this page! asdfasfdasfda

- Eliezer Yudkowsky- Another playpen child
asdf

- Eric Rogstad - Headers demo
Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore …

- Stephanie Zolayvar - Playpen child
asdf

- Erictest Rogstadtest - playpen subpage
playpen subpage clickbait

- Robert Lecnik

- Another playpen child
- Arbital practices
Guidelines and rules for interacting on Arbital.

- Eliezer Yudkowsky- Arbital: Do what works
When deciding things on Arbital, think about the real goals, and move towards them.

- Eric Bruylant- Arbital: Do what works: Justification
**Do what works** is meant to create a firm but flexible foundation for Arbital policy. The aims inc…

- Eric Bruylant

- Arbital: Do what works: Justification

- Arbital: Do what works
- Arbital user groups
Users can attain different powers and responsibilities on Arbital.

- Eric Bruylant- Arbital arbiter
Arbiters provide oversight and dispute resolution to an Arbital domain.

- Eric Bruylant - Arbital reviewer
Reviewers help writers improve their pages, check over all changes to Arbital's content, and assess page quality.

- Eric Bruylant - Arbital trusted user
Trusted users can edit most pages directly, and don't need approval to add pages to a domain.

- Eric Bruylant

- Arbital arbiter
- Contributing to Arbital
Want to help Arbital become awesome?

- Eric Bruylant- Arbital examplar pages
Exemplar pages on Arbital.

- Eric Bruylant - Arbital external resources
Arbital wants to link users to great content, wherever it is!

- Eric Bruylant - Arbital features
Overview of all Arbital features.

- Alexei Andreev- Arbital "parent" relationship
Parent-child relationship between pages implies a strong, inseparable connection.

- Alexei Andreev - Arbital "requires" relationship
A page can require a requisite if the reader needs to have it before they are able to understand the page.

- Alexei Andreev - Arbital "tag" relationship
Tags are a way to connect pages that share a common topic.

- Alexei Andreev - Arbital "teaches" relationship
A page can teach a requisite when the user can acquire it by reading the page.

- Alexei Andreev - Arbital Markdown
All about Arbital's extended Markdown syntax.

- Alexei Andreev- Arbital Markdown questionnaire
How to ask questions in Markdown.

- Alexei Andreev- Questionnaire helper 1
Just a helper page to demonstrate Arbital's questionnaire features.

- Alexei Andreev - Questionnaire helper 2
Just a helper page to demonstrate Arbital's questionnaire features.

- Alexei Andreev - Questionnaire helper 3
Just a helper page to demonstrate Arbital's questionnaire features.

- Alexei Andreev

- Questionnaire helper 1
- Arbital hidden text
How to hide text in Markdown behind a button.

- Alexei Andreev - Arbital markdown demo
Demo of Arbital's markdown

- Eric Bruylant - Arbital page summaries Markdown syntax
How to create page summaries using Arbital's Markdown syntax.

- Alexei Andreev - Arbital todo
So many things todo!

- Eric Bruylant

- Arbital Markdown questionnaire
- Arbital comment
A comment is a way for you to express your thoughts and opinions within the context of a page.

- Alexei Andreev - Arbital content request
Arbital doesn't explain something you'd like to learn? We'd like to know, so we can prioritize.

- Eric Bruylant - Arbital domain
What is a domain? Why is it important?

- Alexei Andreev - Arbital editor
How to use Arbital's page editor.

- Alexei Andreev- Arbital editor buttons
What do all the buttons in the editor do?

- Alexei Andreev - Arbital editor: Advanced
Advanced features of Arbital editor.

- Alexei Andreev - Arbital editor: Basics
The basics of how to use the Arbital editor.

- Alexei Andreev

- Arbital editor buttons
- Arbital greenlink
What happens when you hover over an Arbital link?

- Alexei Andreev - Arbital groups
What are groups? How can I create a new group?

- Alexei Andreev - Arbital lens
A lens is a page that presents another page's content from a different angle.

- Alexei Andreev- Arbital lens: TL;DR
Much shorter version of Arbital Lens page

- Alexei Andreev

- Arbital lens: TL;DR
- Arbital likes
What are likes? When should I use them? What happens when I like something?

- Alexei Andreev - Arbital mark
What is a mark on Arbital? When is it created? Why is it important?

- Alexei Andreev - Arbital page
The Arbital is a series of pages.

- Alexei Andreev- Arbital draft
Drafts are private work-in-progress pages.

- Eric Bruylant - Arbital listed page
Listed pages have been accepted into at least one domain and checked by a reviewer.

- Eric Bruylant - Arbital page alias
The alias is a short, unique name assigned to each page. For example: "arbital_alias". The alias u…

- Eric Rogstad - Arbital page clickbait
The text you are reading right now is clickbait.

- Eric Rogstad - Arbital page title
The title of a page is shown at the top of the page (e.g. "Arbital page title") and in most places …

- Eric Rogstad - Arbital page: Basics
Explaining the basic features of an Arbital page.

- Alexei Andreev - Arbital unlisted page
What do you call a page that's not part of any domain?

- Alexei Andreev

- Arbital draft
- Arbital page summaries
Because only one summary is not enough!

- Alexei Andreev - Arbital path
Arbital path is a linear sequence of pages tailored specifically to teach a given concept to a user.

- Alexei Andreev - Arbital query
What is a query? How to create it? How to resolve it?

- Alexei Andreev - Arbital requisites
To understand a thing you often need to understand some other things.

- Alexei Andreev - Arbital subscriptions
What's a subscription? How do you change it? What to expect?

- Alexei Andreev- Arbital subscriptions: Maintenance
Subscribing to a page with intention of maintaining it.

- Alexei Andreev

- Arbital subscriptions: Maintenance

- Arbital "parent" relationship
- Arbital math levels
How mathy do you like your pages?

- Eric Bruylant - Arbital projects
Arbital projects are small-scale drives to fill in areas of content.

- Eric Bruylant- Arbital completed project
Completed projects are essentially finished, having achieved the goals they set out to.

- Eric Bruylant - Arbital featured project
Featured projects are active, currently promoted projects.

- Eric Bruylant - Arbital ongoing project
Ongoing projects are not complete, but also not high activity enough to promote.

- Eric Bruylant - Arbital project outline
Project outlines are in-progress proposals for projects.

- Eric Bruylant- Project proposal: Complex numbers
Project proposal for complex numbers

- Alexei Andreev

- Project proposal: Complex numbers
- Arbital proposed project
Collecting all project proposals under this page.

- Alexei Andreev- Project outline: Intro to the Universal Property
Outline detailing all the work required for a proposed Arbital Project

- Eric Rogstad

- Project outline: Intro to the Universal Property

- Arbital completed project
- Arbital quality
Arbital's system for tracking page quality.

- Eric Bruylant- Placeholder
This is an empty page created for structural reasons (parent, requisite, or teaches).

- Eric Bruylant - Proposed A-Class
Pages which have been proposed for A-Class status.

- Eric Bruylant

- Placeholder
- Arbital scope
What kind of content is Arbital looking for?

- Eric Bruylant - Author's guide to Arbital
How to write intuitive, flexible content on Arbital.

- Alexei Andreev- Author's guide to Arbital explanations
Requisite used for teaching authors about Arbital explanation features.

- Alexei Andreev - Author's guide to Arbital: Advanced
Requisite used for teaching authors about advanced Arbital features.

- Alexei Andreev - Author's guide to Arbital: Basics
Requisite used for teaching authors about basic Arbital features.

- Alexei Andreev - Author's guide to processing feedback
Requisite used for teaching authors about Arbital feedback features.

- Alexei Andreev - How to author on Arbital!
Want to contribute pages to Arbital? Here's our current version of the ad-hoc guide to being an author!

- Eliezer Yudkowsky

- Author's guide to Arbital explanations
- Possible math pages
A list of things which we may want math pages on

- Eric Bruylant - Style guidelines
Various stylistic conventions people should follow on Arbital

- Alexei Andreev- Link glossary pages for overloaded words
If your subject is using what sound like ordinary-language words in a special sense, create a glossa…

- Eliezer Yudkowsky - Math style guidelines
Stylistic conventions specific to pages about math.

- Dylan Hendrickson - The ideal Arbital math page
Think of the best math textbook you've ever read -- why was it good?

- Eric Rogstad

- Link glossary pages for overloaded words
- Submitting a page to a domain
How and why to submit a page to a domain

- Alexei Andreev

- Arbital examplar pages
- Meta tags
What are meta tags and when to use them?

- Eliezer Yudkowsky- A-Class
This page is well-written, high-quality, and essentially complete.

- Eric Bruylant - B-Class
This page is mostly complete and without major problems, but has not had detailed feedback from the target audience and reviewers.

- Eric Bruylant - C-Class
This page has substantial content, but may not thoroughly cover the topic, may not meet style and prose standards, or may not explain the concept in a way the target audience will reliably understand.

- Eric Bruylant - Concept
Add this meta tag to pages which are concepts.

- Alexei Andreev - Definition
Meta tag used to mark pages that strictly define a particular term or phrase.

- Eliezer Yudkowsky - Disambiguation
Several distinct concepts use this page's name, this page helps readers find what they're looking for.

- Eric Bruylant - Example problem
Tag for pages that provide an example problem referenced by a number of other pages. The summary of…

- Nate Soares - External resources
This lens links out to other great resources across the web.

- Eric Bruylant - Featured
This page is has been selected by Arbital to be featured and promoted.

- Eric Bruylant - Formal definition
This page gives a purely formal definition of a topic, rather than motivating, explaining, and giving examples.

- Eric Bruylant - Formatting issues
This page has formatting or mathjax issues.

- Eric Bruylant - Guide
Meta tag for the start page of a multi-page guide.

- Eric Bruylant - High-speed explanation
Use this tag to indicate that a page offers a relatively faster and more terse explanation. Note th…

- Eric Rogstad - Hub page
This tag is applied to pages which server the role of a "hub": the user starts there, goes off to learn more about the topic, and then comes back. This meta tag modifies the page's UI.

- Alexei Andreev - Image requested
An editor has requested an image for this page.

- Eric Bruylant - Just a requisite
A tag for nodes that just act as part of Arbital's requisite system

- Eliezer Yudkowsky - List
Meta tags for pages that are basically lists.

- Alexei Andreev - Low-speed explanation
Use this tag to indicate that a page offers a relatively slower, more gentle, or more wordy explanat…

- Eric Rogstad - Meta tags which request an edit to the page
Tags that mean your page should be edited.

- Stephanie Zolayvar- Formatting issues
This page has formatting or mathjax issues.

- Eric Bruylant - Image requested
An editor has requested an image for this page.

- Eric Bruylant - Needs accessible summary
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- Formatting issues
- Meta tags which suppress a page from being featured
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- Eric Bruylant - Just a requisite
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- Alexei Andreev - Quality meta tags
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- Alexei Andreev- A-Class
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- A-Class
- More about Arbital
Lots more information about Arbital vision.

- Alexei Andreev- Arbital: Google Maps for knowledge
Take your understanding from where it is to where it wants to be.

- Alexei Andreev - Arbital: better blogging
What makes Arbital the choice blogging platform?

- Alexei Andreev - Arbital: fixing online discussion
How can Arbital do better than existing discussion platforms?

- Alexei Andreev - Arbital: information hub
How will Arbital help you keep up to date on any given subject?

- Alexei Andreev - Arbital: learning from Wikipedia
How is Arbital different from Wikipedia?

- Alexei Andreev - Eliezer's vision for Arbital
Why are we building this? What's the goal?

- Eric Bruylant

- Arbital: Google Maps for knowledge
- Team Arbital
The people behind this site.

- Eric Bruylant - Welcome to Arbital
Front page explaining what Arbital is all about.

- Alexei Andreev

## Arbital Labs

- Discussion norms
What makes conversation productive?

- Eric Rogstad - Meta (Arbital Labs)
Tag for meta discussion in Arbital Labs

- Eric Rogstad

## Arbital as single conversational locus

- Arbital members should state their beliefs clearly, even if they only hold them weakly.
Sometimes there is a tendency for people to state a weak version of their belief, if they don't beli…

- Alexei Andreev - Arbital norms should encourage disagreement. - Alexei Andreev
- Arbital norms should encourage nitpicking. - Alexei Andreev
- Arbital should not require any member to have particular beliefs, but should instead focus on enforcing specific rules of conduct. - Alexei Andreev

## Bone Fracture

- Effect of four monthly oral vitamin D3 (cholecalciferol) supplementation on fractures and mortality in men and women living in the community: randomised double blind controlled trial
Randomised double blind controlled trial. Link: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC150177/…

- Alexei Andreev - The effect of vitamin D supplementation on skeletal, vascular, or cancer outcomes: a trial sequential meta-analysis
Trial sequential meta-analysis. Link: http://www.thelancet.com/journals/landia/article/PIIS2213-858…

- Alexei Andreev - Vitamin D helps prevent bone fracture
(Yes) One randomized trial shows some promise, but at extremely high doses of 100,000 IUs. (No) A l…

- Alexei Andreev

## Calcium

- Calcium Plus Vitamin D Supplementation and the Risk of Breast Cancer
Randomized trial Link: http://jnci.oxfordjournals.org/content/100/22/1581.long From conclusions: …

- Alexei Andreev - Calcium plus Vitamin D Supplementation and the Risk of Colorectal Cancer
Randomized trial Link: http://www.fp.ucalgary.ca/FMResidentSecure/Articles/Ca%20vit%20d%20and%20ca%…

- Alexei Andreev - Vitamin D and calcium supplementation reduces cancer risk: results of a randomized trial
Randomized trial Link: http://www.ncbi.nlm.nih.gov/pubmed/17556697%20 From conclusions: > "Improv…

- Alexei Andreev

## Cancer

- Calcitriol in the Treatment of Prostate Cancer
Review Link: http://ar.iiarjournals.org/content/26/4A/2647.full.pdf From the abstract: > "Calcitr…

- Alexei Andreev - Calcium Plus Vitamin D Supplementation and the Risk of Breast Cancer
Randomized trial Link: http://jnci.oxfordjournals.org/content/100/22/1581.long From conclusions: …

- Alexei Andreev - Calcium plus Vitamin D Supplementation and the Risk of Colorectal Cancer
Randomized trial Link: http://www.fp.ucalgary.ca/FMResidentSecure/Articles/Ca%20vit%20d%20and%20ca%…

- Alexei Andreev - Does solar exposure, as indicated by the non-melanoma skin cancers, protect from solid cancers: Vitamin D as a possible explanation
Correlational study Link: http://www.ejcancer.com/article/S0959-8049%2807%2900324-3/abstract From …

- Alexei Andreev - Molecular basis of the potential of vitamin D to prevent cancer
Theoretical meta-analysis Link: http://www.researchgate.net/publication/5812584_Molecular_basis_of_…

- Alexei Andreev - Serum 25-Hydroxyvitamin D and Risks of Colon and Rectal Cancer in Finnish Men
Case-control study. Link: http://aje.oxfordjournals.org/content/173/5/499.full From discussion: >…

- Alexei Andreev - Serum Vitamin D Concentration and Prostate Cancer Risk: A Nested Case – Control Study
Prospective analysis. Link: http://jnci.oxfordjournals.org/content/100/11/796.full.pdf From conclu…

- Alexei Andreev - The effect of vitamin D supplementation on skeletal, vascular, or cancer outcomes: a trial sequential meta-analysis
Trial sequential meta-analysis. Link: http://www.thelancet.com/journals/landia/article/PIIS2213-858…

- Alexei Andreev - Vitamin D and calcium supplementation reduces cancer risk: results of a randomized trial
Randomized trial Link: http://www.ncbi.nlm.nih.gov/pubmed/17556697%20 From conclusions: > "Improv…

- Alexei Andreev - Vitamin D helps prevent breast cancer
No -- Calcium and vitamin D supplementation did not reduce invasive breast cancer incidence in post…

- Alexei Andreev - Vitamin D helps prevent cancer
There have been a lot of studies performed that show that vitamin D helps prevent cancer, but overal…

- Alexei Andreev - Vitamin D helps prevent colorectal cancer
No -- Daily supplementation of calcium with vitamin D for seven years had no effect on the incidenc…

- Alexei Andreev - Vitamin D helps prevent prostate cancer
Yes --- Calcitriol showed significant antineoplastic activity in pre-clinical models of prostate ca…

- Alexei Andreev - Vitamin, Mineral, and Multivitamin Supplements for the Primary Prevention of Cardiovascular Disease and Cancer: A Systematic Evidence Review for the US Preventive Services Task Force
Meta-study Link: http://www.ncbi.nlm.nih.gov/pubmed/24308073 From results: > "Vitamin D and/or ca…

- Alexei Andreev

## Cardiovascular Disease

- The effect of vitamin D supplementation on skeletal, vascular, or cancer outcomes: a trial sequential meta-analysis
Trial sequential meta-analysis. Link: http://www.thelancet.com/journals/landia/article/PIIS2213-858…

- Alexei Andreev - Vitamin D Deficiency and Risk for Cardiovascular Disease
Theoretical meta-analysis. Link: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2726624/ From conclus…

- Alexei Andreev - Vitamin D helps prevent cardiovascular disease
This question is a bit vague. There are two ways to make it more specific. 1. Does vitamin D deffic…

- Alexei Andreev - Vitamin D is related to blood pressure and other cardiovascular risk factors in middle-aged men
Correlational study Link: http://www.ncbi.nlm.nih.gov/pubmed/8541004

- Alexei Andreev - Vitamin, Mineral, and Multivitamin Supplements for the Primary Prevention of Cardiovascular Disease and Cancer: A Systematic Evidence Review for the US Preventive Services Task Force
Meta-study Link: http://www.ncbi.nlm.nih.gov/pubmed/24308073 From results: > "Vitamin D and/or ca…

- Alexei Andreev

## Central examples

- AIXI
How to build an (evil) superintelligent AI using unlimited computing power and one page of Python code.

- Eliezer Yudkowsky- AIXI-tl
A time-bounded version of the ideal agent AIXI that uses an impossibly large finite computer instead of a hypercomputer.

- Eliezer Yudkowsky

- AIXI-tl
- Happiness maximizer
It is sometimes proposed that we build an AI intended to maximize human happiness. (One early propo…

- Eliezer Yudkowsky

## Claim explainer: donor lotteries and returns to scale

## Convex

- Convex function
A function that only curves upward

- Jessica Taylor - Convex set
A set that contains all line segments between points in the set

- Jessica Taylor

## Crony belief

- Crony beliefs (from Melting Asphalt)
The original article that introduced and explained "merit beliefs" vs "crony beliefs" dichotomy.

- Alexei Andreev

## Decision theory

- Causal decision theories
On CDT, to choose rationally, you should imagine the world where your physical act changes, then imagine running that world forward in time. (Therefore, it's irrational to vote in elections.)

- Eliezer Yudkowsky - Evidential decision theories
Theories which hold that the principle of rational choice is "Choose the act that would be the best news, if somebody told you that you'd chosen that act."

- Eliezer Yudkowsky - Expected utility formalism
Expected utility is the central idea in the quantitative implementation of consequentialism

- Eliezer Yudkowsky- Coherence theorems
A 'coherence theorem' shows that something bad happens to an agent if its decisions can't be viewed as 'coherent' in some sense. E.g., an inconsistent preference ordering leads to going in circles.

- Eliezer Yudkowsky - Coherent decisions imply consistent utilities
Why do we all use the 'expected utility' formalism? Because any behavior that can't be viewed from that perspective, must be qualitatively self-defeating (in various mathy ways).

- Eliezer Yudkowsky - Expected utility
Scoring actions based on the average score of their probable consequences.

- Eliezer Yudkowsky - Expected utility agent
If you're not some kind of expected utility agent, you're going in circles.

- Eliezer Yudkowsky - Utility function
The only coherent way of wanting things is to assign consistent relative scores to outcomes.

- Eliezer Yudkowsky

- Coherence theorems
- Logical decision theories
Root page for topics on logical decision theory, with multiple intros for different audiences.

- Eliezer Yudkowsky- An Introduction to Logical Decision Theory for Everyone Else
So like what the heck is 'logical decision theory' in terms a normal person can understand?

- Eliezer Yudkowsky - Fair problem class
A problem is 'fair' (according to logical decision theory) when only the results matter and not how we get there.

- Eliezer Yudkowsky - Guide to Logical Decision Theory
The entry point for learning about logical decision theory.

- Eliezer Yudkowsky - Introduction to Logical Decision Theory for Analytic Philosophers
Why "choose as if controlling the logical output of your decision algorithm" is the most appealing candidate for the principle of rational choice.

- Eliezer Yudkowsky - Introduction to Logical Decision Theory for Computer Scientists
'Logical decision theory' from a math/programming standpoint, including how two agents with mutual knowledge of each other's code can cooperate on the Prisoner's Dilemma.

- Eliezer Yudkowsky - Introduction to Logical Decision Theory for Economists
An introduction to 'logical decision theory' and its implications for the Ultimatum Game, voting in elections, bargaining problems, and more.

- Eliezer Yudkowsky - Newcomblike decision problems
Decision problems in which your choice correlates with something other than its physical consequences (say, because somebody has predicted you very well) can do weird things to some decision theories.

- Eliezer Yudkowsky- 'Rationality' of voting in elections
"A single vote is very unlikely to swing the election, so your vote is unlikely to have an effect" versus "Many people similar to you are making a similar decision about whether to vote."

- Eliezer Yudkowsky - 99LDT x 1CDT oneshot PD tournament as arguable counterexample to LDT doing better than CDT
Arguendo, if 99 LDT agents and 1 CDT agent are facing off in a one-shot Prisoner's Dilemma tournament, the CDT agent does better on a problem that CDT considers 'fair'.

- Eliezer Yudkowsky - Absent-Minded Driver dilemma
A road contains two identical intersections. An absent-minded driver wants to turn right at the second intersection. "With what probability should the driver turn right?" argue decision theorists.

- Eliezer Yudkowsky - Death in Damascus
Death tells you that It is coming for you tomorrow. You can stay in Damascus or flee to Aleppo. Whichever decision you actually make is the wrong one. This gives some decision theories trouble.

- Eliezer Yudkowsky - Newcomb's Problem
There are two boxes in front of you, Box A and Box B. You can take both boxes, or only Box B. Box A contains $1000. Box B contains $1,000,000 if and only if Omega predicted you'd take only Box B.

- Eliezer Yudkowsky - Parfit's Hitchhiker
You are dying in the desert. A truck-driver who is very good at reading faces finds you, and offers to drive you into the city if you promise to pay $1,000 on arrival. You are a selfish rationalist.

- Eliezer Yudkowsky - Prisoner's Dilemma
You and an accomplice have been arrested. Both of you must decide, in isolation, whether to testify against the other prisoner--which subtracts one year from your sentence, and adds two to theirs.

- Eliezer Yudkowsky- True Prisoner's Dilemma
A scenario that would reproduce the ideal payoff matrix of the Prisoner's Dilemma about human beings who care about their public reputation and each other.

- Eliezer Yudkowsky

- True Prisoner's Dilemma
- Toxoplasmosis dilemma
A parasitic infection, carried by cats, may make humans enjoy petting cats more. A kitten, now in front of you, isn't infected. But if you *want* to pet it, you may already be infected. Do you?

- Eliezer Yudkowsky - Transparent Newcomb's Problem
Omega has left behind a transparent Box A containing $1000, and a transparent Box B containing $1,000,000 or nothing. Box B is full iff Omega thinks you one-box on seeing a full Box B.

- Eliezer Yudkowsky - Ultimatum Game
A Proposer decides how to split $10 between themselves and the Responder. The Responder can take what is offered, or refuse, in which case both parties get nothing.

- Eliezer Yudkowsky

- 'Rationality' of voting in elections
- Omega (alien philosopher-troll)
The entity that sets up all those trolley problems. An alien philosopher/troll imbued with unlimited powers, excellent predictive ability, and very odd motives.

- Eliezer Yudkowsky - Updateless decision theories
Decision theories that maximize their policies (mappings from sense inputs to actions), rather than using their sense inputs to update their beliefs and then selecting actions.

- Eliezer Yudkowsky

- An Introduction to Logical Decision Theory for Everyone Else
- Modal combat
Modal combat

- Jaime Sevilla Molina

## Early-life planning

- An early stage prioritisation model
How do you choose which projects to work on, early on in life?

- Ben Pace - An early stage project generation model
How do you figure out which projects to even consider, when you're getting started?

- Ben Pace - Open questions in life-planning
What questions do the two models above not solve?

- Ben Pace

## Effective altruism

- Donor coordination
How should donors behave, given that their actions may affect other donors?

- Eric Rogstad - Existential risk
> Because of accelerating technological progress, humankind may be rapidly approaching a critical ph…

- Eric Rogstad

## Eliezer Yudkowsky

- How to build your own Lumenator
Treating Seasonal Affective Disorder using MOAR LIGHT can sometimes solve what dinky little lightboxes can't.

- Eliezer Yudkowsky

## Eliezer office hours

- Eliezer office hours 2016-04-19
### Steph's notes from the actual meeting: - "good question" -> people upvoting your question is a g…

- Eric Rogstad - Eliezer office hours 2016-05-07
##Questions for Eliezer## ###Strategy### Q: What does a discussion of which EA cause to donate to …

- Eric Rogstad

## Eric Bruylant

- Browser optimization for autodidacts
Your browser is your window on the 'net. Make it awesome.

- Eric Bruylant

## Geneva Macro Labs

- Solutions Paper - Solving Employment in the Light of Automation
GeM Labs solution space to tackle labour market challenges that arise from the rise of automation and artificial intelligence.

- Geneva Macro Labs

## Geometric algebra

- Combining vectors
One of the most useful things we can do with vectors is to combine them!

- Adele Lopez - Geometric product
#Motivation want to incorporate rotors like $e^{\text{I}\theta}$ and scalars $n$ in the same system …

- Adele Lopez- Geometric product: summary
Product for [multivectors multivectors]. Associative, left and right distributive. Non-commutative. …

- Adele Lopez

- Geometric product: summary
- Geometric product: summary
Product for [multivectors multivectors]. Associative, left and right distributive. Non-commutative. …

- Adele Lopez - Geometry of vectors: direction
What rotation would it take to line up this vector to this one?

- Adele Lopez - Vector arithmetic
Vectors: what they are, and how to add and scale them.

- Adele Lopez

## Jaime Sevilla Molina

- Teleological Measure and agency
A crackpot's research proposal

- Jaime Sevilla Molina

## Joint probability distribution

- Joint probability distribution: (Motivation) coherent probabilities
If you don't use joint probability distributions, none of your probabilities will make any sense. So, yeah, use joint probability distributions.

- Tsvi BT

## Logic

- Arguments
An argument is a formal reasoning, valid or not.

- Jeremy Perret - Conjunctions and disjunctions
The fancy name for the "and" and "or" connectives.

- Jeremy Perret - Formal Logic
Formal logic studies the form of correct arguments through rigorous and precise mathematical theories.

- Erik Istre- Consistency
A consistent [-theory] is one in which there are well-formed statements that you cannot prove from i…

- Jaime Sevilla Molina - Logical system
Logical systems (a.k.a. formal systems) are mathematical abstractions that aim to capture the notion…

- Jaime Sevilla Molina

- Consistency
- Law of syllogism
Deriving something from the conclusion of another thing.

- Jeremy Perret - Logical system
Logical systems (a.k.a. formal systems) are mathematical abstractions that aim to capture the notion…

- Jaime Sevilla Molina - Modus tollens
Deriving a negation from another negation

- Jeremy Perret - Negation of propositions
The proposition that is false if another one is true and vice-versa.

- Jeremy Perret - Propositions
Propositions are statements with a truth value.

- Jeremy Perret

## Market monetarism

- NGDP level targeting
Central banks ought to regularize the total flow of money to increase at a predictable 5% rate per year, and doing this would solve a surprising number of other problems.

- Eliezer Yudkowsky- Path targeting
Don't say "We want this price to go up at 2%/year", say "We want this to be $1 in year 1, $1.02 in year 2, $1.04 in year 3" and don't change the rest of the path if you miss one year's target.

- Eliezer Yudkowsky

- Path targeting

## Mathematics

- Ability to read algebra
Do you have sufficient mathematical ability that you can read a sentence that uses some algebra or invokes a mathematical idea, without slowing down too much?

- Eliezer Yudkowsky - Ability to read calculus
Can you take integral signs and differentiations in stride?

- Eliezer Yudkowsky - Ability to read logic
Can you read sentences symbolically stating "For all x: exists y: phi(x, y) or not theta(y)" without slowing down too much?

- Eliezer Yudkowsky - Abstract algebra
The study of groups, fields, vector spaces, arithmetics, algebras, and more.

- Nate Soares- Algebraic structure
Roughly speaking, an algebraic structure is a set $X$, known as the underlying set, paired with a co…

- Nate Soares- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Nate Soares - Algebraic structure tree
When is a monoid a semilattice? What's the difference between a semigroup and a groupoid? Find out here!

- Ryan Hendrickson - Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

- Nate Soares- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Nate Soares - Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).

- Patrick Stevens- Alternating group is generated by its three-cycles
A useful result which lets us prove things about the alternating group more easily.

- Patrick Stevens - Conjugacy classes of the alternating group on five elements
$A_5$ has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.

- Patrick Stevens- Conjugacy classes of the alternating group on five elements: Simpler proof
A listing of the conjugacy classes of the alternating group on five letters, without using heavy theory.

- Patrick Stevens

- Conjugacy classes of the alternating group on five elements: Simpler proof
- Splitting conjugacy classes in alternating group
The conjugacy classes in the alternating group are usually the same as those in the symmetric group; there is a surprisingly simple condition for when this does not hold.

- Patrick Stevens - The alternating group on five elements is simple
The smallest (nontrivial) simple group is the alternating group on five elements.

- Patrick Stevens- The alternating group on five elements is simple: Simpler proof
A proof which avoids some of the heavy machinery of the main proof.

- Patrick Stevens

- The alternating group on five elements is simple: Simpler proof
- The alternating groups on more than four letters are simple
The alternating groups are the most accessible examples of simple groups, and this fact also tells us that the symmetric groups are "complicated" in some sense.

- Patrick Stevens - The collection of even-signed permutations is a group
This proves the well-definedness of one particular definition of the alternating group.

- Patrick Stevens

- Alternating group is generated by its three-cycles
- Cauchy's theorem on subgroup existence
Cauchy's theorem is a useful condition for the existence of cyclic subgroups of finite groups.

- Patrick Stevens- Cauchy's theorem on subgroup existence: intuitive version
Cauchy's Theorem states that if $G$ is a finite [-group], and $p$ is a prime dividing the order of $…

- Patrick Stevens

- Cauchy's theorem on subgroup existence: intuitive version
- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Mark Chimes - Cyclic group
Cyclic groups form one of the most simple classes of groups.

- Patrick Stevens- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Mark Chimes

- Cyclic Group Intro (Math 0)
- Dihedral group
The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.

- Patrick Stevens- Dihedral groups are non-abelian
The group of symmetries of the triangle and all larger regular polyhedra are not abelian.

- Patrick Stevens

- Dihedral groups are non-abelian
- Every group is a quotient of a free group
Given a group $G$, there is a Free group $F(X)$ on some set $X$, such that $G$ is isomorphic to some…

- Patrick Stevens - Free group
The free group is "the purest way to make a group containing a given set".

- Patrick Stevens- Formal definition of the free group
Van der Waerden's trick lets us define the free groups in a slick but mostly incomprehensible way.

- Patrick Stevens - Free group universal property
The Free group may be defined by a Universal property, allowing Category theory to talk about free …

- Patrick Stevens - Free groups are torsion-free
An easy way to determine that many groups are not free: free groups contain no non-identity elements of finite order.

- Patrick Stevens

- Formal definition of the free group
- Group conjugate
Conjugation lets us perform permutations "from the point of view of" another permutation.

- Patrick Stevens - Group coset
Given a subgroup $H$ of Group $G$, the *left cosets* of $H$ in $G$ are sets of the form $\{ gh : h \…

- Patrick Stevens- Left cosets are all in bijection
The left cosets of a subgroup in a parent group are all the same size.

- Patrick Stevens - Left cosets partition the parent group
In a group, every element has a unique coset in which it lies, allowing us to compress some of the information about the group.

- Patrick Stevens

- Left cosets are all in bijection
- Group homomorphism
A group homomorphism is a "function between groups" that "respects the group structure".

- Patrick Stevens- Image of the identity under a group homomorphism is the identity
All group homomorphisms preserve the identity.

- Patrick Stevens - Kernel of group homomorphism
The kernel of a Group homomorphism $f: G \to H$ is the collection of all elements $g$ in $G$ such th…

- Patrick Stevens - The composition of two group homomorphisms is a homomorphism
The collection of group homomorphisms is closed under composition.

- Patrick Stevens - The image of a group under a homomorphism is a subgroup of the codomain
Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.

- Patrick Stevens - Under a group homomorphism, the image of the inverse is the inverse of the image
The operations of "taking inverses" and "applying a group homomorphism" commute: it does not matter in which order we do them.

- Patrick Stevens

- Image of the identity under a group homomorphism is the identity
- Group isomorphism
"Isomorphism" is the proper notion of "sameness" or "equality" among groups.

- Patrick Stevens - Group orbit
When we have a group acting on a set, we are often interested in how the group acts on a particular …

- Adele Lopez - Group presentation
Presentations are a fairly compact way of expressing groups.

- Patrick Stevens - Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?

- Qiaochu Yuan - Group: Exercises
Test your understanding of the definition of a group with these exercises.

- Qiaochu Yuan - Lagrange theorem on subgroup size
Lagrange's Theorem is an important restriction on the sizes of subgroups of a finite group.

- Patrick Stevens- Lagrange theorem on subgroup size: Intuitive version
Lagrange's theorem strongly restricts the size a subgroup of a group can be.

- Patrick Stevens

- Lagrange theorem on subgroup size: Intuitive version
- Normal subgroup
Normal subgroups are subgroups which are in some sense "the same from all points of view".

- Patrick Stevens- Quotient by subgroup is well defined if and only if subgroup is normal
Let $G$ be a Group and $N$ a Normal subgroup of $G$. Then we may define the *quotient group* $G/N$ t…

- Patrick Stevens - Subgroup is normal if and only if it is a union of conjugacy classes
A useful way to think about normal subgroups, which meshes with their "closed under conjugation" interpretation.

- Patrick Stevens - Subgroup is normal if and only if it is the kernel of a homomorphism
The "correct way" to think about normal subgroups is as kernels of homomorphisms.

- Patrick Stevens

- Quotient by subgroup is well defined if and only if subgroup is normal
- Order of a group
The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ an…

- Nate Soares - Order of a group element
Given an element $g$ of group $(G, +)$ (which henceforth we abbreviate simply as $G$), the order of …

- Patrick Stevens - Prime order groups are cyclic
This is the first step on the road to classifying the finite groups.

- Patrick Stevens - Simple group
The simple groups form the "building blocks" of group theory, analogously to the prime numbers in number theory.

- Patrick Stevens - Subgroup
A group that lives inside a bigger group.

- Dylan Hendrickson - Symmetric group
The symmetric groups form the fundamental link between group theory and the notion of symmetry.

- Patrick Stevens- Cayley's Theorem on symmetric groups
The "fundamental theorem of symmetry", forging the connection between symmetry and group theory.

- Patrick Stevens - Conjugacy class is cycle type in symmetric group
There is a neat characterisation of the conjugacy classes in the symmetric group on a finite set.

- Patrick Stevens - Conjugacy classes of the symmetric group on five elements
The symmetric group on five elements is a group of just the right size to make a good example of a table of conjugacy classes.

- Patrick Stevens - Cycle notation in symmetric groups
Cycle notation is a convenient way to represent the elements of a symmetric group.

- Patrick Stevens- Cycle type of a permutation
The cycle type is an invariant of a permutation in the symmetric group.

- Patrick Stevens - Disjoint cycle notation is unique
Disjoint cycle notation provides a canonical way to express elements of the symmetric group.

- Patrick Stevens

- Cycle type of a permutation
- Disjoint cycles commute in symmetric groups
In cycle notation, if two cycles are disjoint, then they commute.

- Patrick Stevens - Every member of a symmetric group on finitely many elements is a product of transpositions
This fact can often simplify arguments about permutations: if we can show that something holds for transpositions, and that it holds for products, then it holds for everything.

- Patrick Stevens - Sign homomorphism (from the symmetric group)
The sign homomorphism is how we extract the alternating group from the symmetric group.

- Patrick Stevens - The sign of a permutation is well-defined
This result is what allows the alternating group to exist.

- Patrick Stevens - Transposition (as an element of a symmetric group)
A transposition is the simplest kind of permutation: it swaps two elements.

- Patrick Stevens

- Cayley's Theorem on symmetric groups

- Abelian group
- Monoid
A monoid $M$ is a pair $(X, \diamond)$ where $X$ is a [set\_theory\_set set] and $\diamond$ is an [a…

- Nate Soares - Ring
A ring is a kind of Algebraic structure which we obtain by considering groups as being "things with…

- Nate Soares- Euclidean domains are principal ideal domains
A Euclidean domain is one where we may somehow perform the division algorithm; this gives us access to some of the nicest properties of the integers.

- Patrick Stevens - Ideals are the same thing as kernels of ring homomorphisms
In ring theory, the notion of "[ideal\_ring\_theory ideal]" corresponds precisely with the notion o…

- Patrick Stevens - In a principal ideal domain, "prime" and "irreducible" are the same
Principal ideal domains have a very useful property that we don't need to distinguish between the informal notion of "prime" (i.e. "irreducible") and the formal notion.

- Patrick Stevens - Integral domain
An integral domain is a ring where the only way to express zero as a product is by having zero as one of the terms.

- Patrick Stevens - Irreducible element (ring theory)
This is the appropriate abstraction of the concept of "prime number" to general rings.

- Patrick Stevens - Kernel of ring homomorphism
The kernel of a ring homomorphism is the collection of things which that homomorphism sends to 0.

- Patrick Stevens - Ordered ring
A ring with a total ordering compatible with its ring structure.

- Dylan Hendrickson - Prime element of a ring
Despite the name, "prime" in ring theory refers not to elements which are "multiplicatively irreducible" but to those such that if they divide a product then they divide some term of the product.

- Patrick Stevens - Principal ideal domain
A principal ideal domain is a kind of ring, in which all ideals have a certain nice form.

- Patrick Stevens - Unique factorisation domain
This is the correct way to abstract from the integers the fact that every integer can be written uniquely as a product of prime numbers.

- Patrick Stevens - Unit (ring theory)
A unit in a ring is just an element with a multiplicative inverse.

- Patrick Stevens

- Euclidean domains are principal ideal domains

- Abelian group
- Underlying set
What do a Group, a Partially ordered set, and a [ topological space] have in common? Each is a Set …

- Nate Soares

- Algebraic structure
- An introductory guide to modern logic
Logic, provability, Löb, Gödel and more!

- Jaime Sevilla Molina - Arithmetical hierarchy
The arithmetical hierarchy is a way of classifying logical statements by the number of clauses saying "for every object" and "there exists an object".

- Eliezer Yudkowsky- Arithmetical hierarchy: If you don't read logic
The arithmetical hierarchy is a way of stratifying statements by how many "for every number" and "th…

- Eliezer Yudkowsky

- Arithmetical hierarchy: If you don't read logic
- Associative operation
An **associative operation** $\bullet : X \times X \to X$ is a binary operation such that for all $x…

- Nate Soares- Associativity vs commutativity
Associativity and commutativity are often confused, because they are both constraints on how a funct…

- Nate Soares - Associativity: Examples
Yes: [Addition], [multiplication], string concatenation. No: [subtraction], [division], a Function …

- Nate Soares - Associativity: Intuition
Associative functions can be interpreted as families of functions that reduce lists down to a singl…

- Nate Soares - Generalized associative law
Given an associative operator $\cdot$ and a list $[a, b, c, \ldots]$ of parameters, all ways of red…

- Nate Soares

- Associativity vs commutativity
- Axiom
An **axiom** of a [theory\_mathematics theory] $T$ is a [well\_formed well-formed] [sentence\_mathem…

- Eric Bruylant- Axiom of Choice
The most controversial axiom of the 20th century.

- Mark Chimes- Axiom of Choice: Definition (Formal)
Mathematically speaking, what does the Axiom of Choice say?

- Mark Chimes- Axiom of Choice Definition (Intuitive)
Definition of the Axiom of Choice, without using heavy mathematical notation.

- Mark Chimes

- Axiom of Choice Definition (Intuitive)
- Axiom of Choice Definition (Intuitive)
Definition of the Axiom of Choice, without using heavy mathematical notation.

- Mark Chimes - Axiom of Choice: Guide
Learn about the most controversial axiom of the 20th century.

- Mark Chimes - Axiom of Choice: History and Controversy
Really? The *most* controversial axiom of the 20th century? Yes.

- Mark Chimes - Axiom of Choice: Introduction
What is the axiom of choice and why do I care?

- Mark Chimes

- Axiom of Choice: Definition (Formal)

- Axiom of Choice
- Binary notation
A way to write down numbers using powers of two.

- Malcolm McCrimmon - Bit
The term "bit" refers to different concepts in different fields. The common theme across all the us…

- Nate Soares- Bit (abstract)
An abstract bit is an element of the set $\mathbb B$, which has two elements. An abstract bit is to …

- Nate Soares - Bit (of data)
A bit of data is the amount of data required to single out one message from a set of two. Equivalen…

- Nate Soares- Bit (of data): Examples
In the game "20 questions", one player (the "leader") thinks of a concept, and the other players ask…

- Nate Soares - Fractional bits
It takes $\log_2(8) = 3$ bits of data to carry one message from a set of 8 possible messages. Simila…

- Nate Soares- Fractional bits: Expected cost interpretation
In the GalCom thought experiment, you regularly have to send large volumes of information through de…

- Nate Soares

- Fractional bits: Expected cost interpretation
- How many bits to a trit?
$\log_2(3) \approx 1.585.$ This can be interpreted a few different ways: 1. If you multiply the nu…

- Nate Soares- Encoding trits with GalCom bits
There are $\log_2(3) \approx 1.585$ bits to a Trit. Why is it that particular value? Consider the Ga…

- Nate Soares

- Encoding trits with GalCom bits

- Bit (of data): Examples
- Shannon
The shannon (Sh) is a unit of Information. One shannon is the difference in [info\_entropy entropy] …

- Nate Soares

- Bit (abstract)
- Bit (abstract)
An abstract bit is an element of the set $\mathbb B$, which has two elements. An abstract bit is to …

- Nate Soares - Boolean
A value in logic that evaluates to either "true" or "false".

- Malcolm McCrimmon - Category theory
How mathematical objects are related to others in the same category.

- Mark Chimes- Category (mathematics)
A description of how a collection of mathematical objects are related to one another.

- Mark Chimes - Category of finite sets
The category of finite sets is exactly what it claims to be. It's a useful training ground for some of the ideas of category theory.

- Patrick Stevens - Equaliser (category theory)
In Category theory, an *equaliser* of a pair of arrows $f, g: A \to B$ is an object $E$ and a univer…

- Patrick Stevens - Object identity via interactions
If we think of objects as opaque "black boxes", how can we tell whether two objects are different? By looking at how they interact with other objects!

- Patrick Stevens - Product (Category Theory)
How a product is characterized rather than how it's constructed

- Mark Chimes- Universal property of the product
The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

- Patrick Stevens- Product is unique up to isomorphism
If something satisfies the universal property of the product, then it is uniquely specified by that property, up to isomorphism.

- Patrick Stevens

- Product is unique up to isomorphism

- Universal property of the product
- Universal property
A universal property is a way of defining an object based purely on how it interacts with other objects, rather than by any internal property of the object itself.

- Patrick Stevens- Universal property of the disjoint union
Just as the empty set may be described by a universal property, so too may the disjoint union of sets.

- Patrick Stevens - Universal property of the empty set
The empty set can be characterised by how it interacts with other sets, rather than by any explicit property of the empty set itself.

- Patrick Stevens- The empty set is the only set which satisfies the universal property of the empty set
This theorem tells us that the universal property provides a sensible way to define the empty set uniquely.

- Patrick Stevens

- The empty set is the only set which satisfies the universal property of the empty set
- Universal property of the product
The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

- Patrick Stevens- Product is unique up to isomorphism
If something satisfies the universal property of the product, then it is uniquely specified by that property, up to isomorphism.

- Patrick Stevens

- Product is unique up to isomorphism

- Universal property of the disjoint union

- Category (mathematics)
- Church-Turing thesis
A thesis about computational models

- Jaime Sevilla Molina- Church-Turing thesis: Evidence for the Church-Turing thesis
Why do we believe in CT thesis?

- Jaime Sevilla Molina - Strong Church Turing thesis
A strengthening of the Church Turing thesis

- Jaime Sevilla Molina

- Church-Turing thesis: Evidence for the Church-Turing thesis
- Closure
A set $S$ is _closed_ under an operation $f$ if, whenever $f$ is fed elements of $S$, it produces an…

- Nate Soares - Colon-to notation
Find out what the notation "f : X -> Y" means that everyone keeps using.

- Qiaochu Yuan - Commutative operation
A commutative function $f$ is a function that takes multiple inputs from a set $X$ and produces an o…

- Nate Soares- Associativity vs commutativity
Associativity and commutativity are often confused, because they are both constraints on how a funct…

- Nate Soares - Commutativity: Examples
Yes: addition, multiplication, maximum, minimum, rock-paper-scissors. No: subtraction, division, st…

- Nate Soares - Commutativity: Intuition
We can think of commutativity either as an artifact of notation, or as a symmetry in the output of a…

- Nate Soares

- Associativity vs commutativity
- Complexity theory
Study of the computational resources needed to compute something

- Jaime Sevilla Molina- Complexity theory: Complexity zoo
Pass and see the exotic beasts coming from the lands of afar!

- Jaime Sevilla Molina - Decision problem
Formalization of general problems

- Jaime Sevilla Molina - P (Polynomial Time Complexity Class)
P is the class of problems which can be solved by algorithms whose run time is bounded by a polynomial.

- Eric Leese - P vs NP
Is creativity purely mechanical?

- Jaime Sevilla Molina- P vs NP: Arguments against P=NP
Why we believe P and NP are different

- Jaime Sevilla Molina

- P vs NP: Arguments against P=NP

- Complexity theory: Complexity zoo
- Computer Programming Familiarity
Want to see programming analogies and applications in your math explanations? Mark this as known.

- Kevin Clancy - Conjugacy class
In a group, the elements can be partitioned naturally into certain classes.

- Patrick Stevens - Consistency
A consistent [-theory] is one in which there are well-formed statements that you cannot prove from i…

- Jaime Sevilla Molina - Countability
Some infinities are bigger than others. Countable infinities are the smallest infinities.

- Alexei Andreev - Currying
Transforms a function of many arguments into a function into a function of a single argument

- M Yass - Decimal notation
The winning architecture for numerals

- Michael Cohen- 0.999...=1
No, it's not "infinitesimally far" from 1 or anything like that. 0.999... and 1 are literally the same number.

- Dylan Hendrickson

- 0.999...=1
- Decit
Decimal digit

- Nate Soares - Derivative
How things change

- Michael Cohen - Diagonal lemma
Constructing self-referential sentences

- Jaime Sevilla Molina- Quine
A computer program that prints (or does other computations to) its own source code, using indirect self-reference.

- Patrick LaVictoire

- Quine
- Digit wheel
A mechanical device for storing a number from 0 to 9. ![](http://www.cl.cam.ac.uk/~djg11/howcompu…

- Nate Soares - Element
**Placeholder**

- Eric Bruylant- Identity element
An element in a set with a binary operation that leaves every element unchanged when used as the other operand.

- Joe Zeng

- Identity element
- Elementary Algebra
How do we describe relations between different things? How can we figure out new true things from tr…

- Adele Lopez - Emulating digits
In general, given enough $n$-digits, you can emulate an $m$-digit, for any $m, n \in$ $\mathbb N$. I…

- Nate Soares - Euclid's Lemma on prime numbers
A very basic but vitally important property of the prime numbers is that they "can't be split between factors": if a prime divides a product then it must divide one of the individual factors.

- Patrick Stevens - Examination through isomorphism
Isomorphism is the correct notion of equality between objects in a category. From the category-theor…

- Luke Sciarappa - Expected value
Trying to assign value to an uncertain state? The weighted average of outcomes is probably the tool you need.

- Michael Cohen - Exponential notation for function spaces
Why $Y^X$ is good notation for the space of maps from $X$ to $Y$

- Izaak Meckler - Factorial
The number of ways you can order things. (Alternately subtitled: Is that exclamation point a factorial, or are you just excited to see me?)

- Michael Cohen- Factorial
The *factorial* of a number $n$ is how we describe "how many different ways we can arrange $n$ obje…

- Patrick Stevens

- Factorial
- Featured math content
Some Arbital pages we think are great!

- Eric Bruylant - Freely reduced word
"Freely reduced" captures the idea of "no cancellation" in a free group.

- Patrick Stevens - Function
Intuitively, a function $f$ is a procedure (or machine) that takes an input and performs some opera…

- Nate Soares- Ackermann function
The slowest-growing fast-growing function.

- Alex Appel - Arity (of a function)
The arity of a function is the number of parameters that it takes. For example, the function $f(a, b…

- Nate Soares - Bijective function
A bijective function is a function with an inverse.

- Patrick Stevens- Bijective Function: Intro (Math 0)
Two boxes are bijective if they contain the same number of items.

- Mark Chimes

- Bijective Function: Intro (Math 0)
- Binary function
A binary function $f$ is a function of two inputs (i.e., a function with arity 2). For example, $+,$…

- Nate Soares - Ceiling
The ceiling of a real number $x,$ denoted $\lceil x \rceil$ or sometimes $\operatorname{ceil}(x),$ i…

- Nate Soares - Codomain (of a function)
The codomain $\operatorname{cod}(f)$ of a function $f : X \to Y$ is $Y$, the set of possible outputs…

- Nate Soares- Codomain vs image
It is useful to distinguish codomain from image both (a) when the type of thing that the function pr…

- Nate Soares

- Codomain vs image
- Codomain vs image
It is useful to distinguish codomain from image both (a) when the type of thing that the function pr…

- Nate Soares - Convex function
A function that only curves upward

- Jessica Taylor - Currying
Transforms a function of many arguments into a function into a function of a single argument

- M Yass - Domain (of a function)
The domain $\operatorname{dom}(f)$ of a function $f : X \to Y$ is $X$, the set of valid inputs for t…

- Nate Soares - Exponential
Any function that constantly gets larger as a proportion of itself.

- Joe Zeng - Function: Physical metaphor
Many functions can be visualized as physical mechanisms of wheels and gears, that take their inputs …

- Nate Soares - Image (of a function)
The image $\operatorname{im}(f)$ of a function $f : X \to Y$ is the set of all possible outputs of $…

- Nate Soares- Codomain vs image
It is useful to distinguish codomain from image both (a) when the type of thing that the function pr…

- Nate Soares

- Codomain vs image
- Injective function
A Function $f: X \to Y$ is *injective* if it has the property that whenever $f(x) = f(y)$, it is the…

- Patrick Stevens - Inverse function
The inverse of a function returns an input of the original function when fed the original's corresponding output.

- Michael Cohen - Operator
An operation $f$ on a set $S$ is a function that takes some values from $S$ and produces a new value…

- Nate Soares - Partial function
A partial function is one which "might not be defined everywhere one might expect it to be".

- Patrick Stevens - Range (of a function)
The "range" of a function is an ambiguous term that is generally used to refer to either the functio…

- Nate Soares - Surjective function
A surjective function is one which "hits everything in the codomain".

- Patrick Stevens - concat (function)
The string concatenation function `concat` puts two strings together, i.e., `concat("one","two")="on…

- Nate Soares

- Ackermann function
- Fundamental Theorem of Arithmetic
The FTA tells us that natural numbers can be decomposed uniquely into prime factors; it is the basis of almost all number theory.

- Patrick Stevens - Generalized element
A category-theoretic generalization of the notion of element of a set.

- Luke Sciarappa - Greatest common divisor
The greatest common divisor of two natural numbers is… the largest number which is a divisor of both. The clue is in the name, really.

- Patrick Stevens- Bézout's theorem
Bézout's theorem is an important link between highest common factors and the integer solutions of a certain equation.

- Patrick Stevens

- Bézout's theorem
- Group theory
What kinds of symmetry can an object have?

- Nate Soares- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

- Nate Soares- Abelian group
A group where the operation commutes. Named after Niels Henrik Abel.

- Nate Soares - Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).

- Patrick Stevens- Alternating group is generated by its three-cycles
A useful result which lets us prove things about the alternating group more easily.

- Patrick Stevens - Conjugacy classes of the alternating group on five elements
$A_5$ has easily-characterised conjugacy classes, based on a rather surprising theorem about when conjugacy classes in the symmetric group split.

- Patrick Stevens- Conjugacy classes of the alternating group on five elements: Simpler proof
A listing of the conjugacy classes of the alternating group on five letters, without using heavy theory.

- Patrick Stevens

- Conjugacy classes of the alternating group on five elements: Simpler proof
- Splitting conjugacy classes in alternating group
The conjugacy classes in the alternating group are usually the same as those in the symmetric group; there is a surprisingly simple condition for when this does not hold.

- Patrick Stevens - The alternating group on five elements is simple
The smallest (nontrivial) simple group is the alternating group on five elements.

- Patrick Stevens- The alternating group on five elements is simple: Simpler proof
A proof which avoids some of the heavy machinery of the main proof.

- Patrick Stevens

- The alternating group on five elements is simple: Simpler proof
- The alternating groups on more than four letters are simple
The alternating groups are the most accessible examples of simple groups, and this fact also tells us that the symmetric groups are "complicated" in some sense.

- Patrick Stevens - The collection of even-signed permutations is a group
This proves the well-definedness of one particular definition of the alternating group.

- Patrick Stevens

- Alternating group is generated by its three-cycles
- Cauchy's theorem on subgroup existence
Cauchy's theorem is a useful condition for the existence of cyclic subgroups of finite groups.

- Patrick Stevens- Cauchy's theorem on subgroup existence: intuitive version
Cauchy's Theorem states that if $G$ is a finite [-group], and $p$ is a prime dividing the order of $…

- Patrick Stevens

- Cauchy's theorem on subgroup existence: intuitive version
- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Mark Chimes - Cyclic group
Cyclic groups form one of the most simple classes of groups.

- Patrick Stevens- Cyclic Group Intro (Math 0)
A finite cyclic group is a little bit like a clock.

- Mark Chimes

- Cyclic Group Intro (Math 0)
- Dihedral group
The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.

- Patrick Stevens- Dihedral groups are non-abelian
The group of symmetries of the triangle and all larger regular polyhedra are not abelian.

- Patrick Stevens

- Dihedral groups are non-abelian
- Every group is a quotient of a free group
Given a group $G$, there is a Free group $F(X)$ on some set $X$, such that $G$ is isomorphic to some…

- Patrick Stevens - Free group
The free group is "the purest way to make a group containing a given set".

- Patrick Stevens- Formal definition of the free group
Van der Waerden's trick lets us define the free groups in a slick but mostly incomprehensible way.

- Patrick Stevens - Free group universal property
The Free group may be defined by a Universal property, allowing Category theory to talk about free …

- Patrick Stevens - Free groups are torsion-free
An easy way to determine that many groups are not free: free groups contain no non-identity elements of finite order.

- Patrick Stevens

- Formal definition of the free group
- Group conjugate
Conjugation lets us perform permutations "from the point of view of" another permutation.

- Patrick Stevens - Group coset
Given a subgroup $H$ of Group $G$, the *left cosets* of $H$ in $G$ are sets of the form $\{ gh : h \…

- Patrick Stevens- Left cosets are all in bijection
The left cosets of a subgroup in a parent group are all the same size.

- Patrick Stevens - Left cosets partition the parent group
In a group, every element has a unique coset in which it lies, allowing us to compress some of the information about the group.

- Patrick Stevens

- Left cosets are all in bijection
- Group homomorphism
A group homomorphism is a "function between groups" that "respects the group structure".

- Patrick Stevens- Image of the identity under a group homomorphism is the identity
All group homomorphisms preserve the identity.

- Patrick Stevens - Kernel of group homomorphism
The kernel of a Group homomorphism $f: G \to H$ is the collection of all elements $g$ in $G$ such th…

- Patrick Stevens - The composition of two group homomorphisms is a homomorphism
The collection of group homomorphisms is closed under composition.

- Patrick Stevens - The image of a group under a homomorphism is a subgroup of the codomain
Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.

- Patrick Stevens - Under a group homomorphism, the image of the inverse is the inverse of the image
The operations of "taking inverses" and "applying a group homomorphism" commute: it does not matter in which order we do them.

- Patrick Stevens

- Image of the identity under a group homomorphism is the identity
- Group isomorphism
"Isomorphism" is the proper notion of "sameness" or "equality" among groups.

- Patrick Stevens - Group orbit
When we have a group acting on a set, we are often interested in how the group acts on a particular …

- Adele Lopez - Group presentation
Presentations are a fairly compact way of expressing groups.

- Patrick Stevens - Group: Examples
Why would anyone have invented groups, anyway? What were the historically motivating examples, and what examples are important today?

- Qiaochu Yuan - Group: Exercises
Test your understanding of the definition of a group with these exercises.

- Qiaochu Yuan - Lagrange theorem on subgroup size
Lagrange's Theorem is an important restriction on the sizes of subgroups of a finite group.

- Patrick Stevens- Lagrange theorem on subgroup size: Intuitive version
Lagrange's theorem strongly restricts the size a subgroup of a group can be.

- Patrick Stevens

- Lagrange theorem on subgroup size: Intuitive version
- Normal subgroup
Normal subgroups are subgroups which are in some sense "the same from all points of view".

- Patrick Stevens- Quotient by subgroup is well defined if and only if subgroup is normal
Let $G$ be a Group and $N$ a Normal subgroup of $G$. Then we may define the *quotient group* $G/N$ t…

- Patrick Stevens - Subgroup is normal if and only if it is a union of conjugacy classes
A useful way to think about normal subgroups, which meshes with their "closed under conjugation" interpretation.

- Patrick Stevens - Subgroup is normal if and only if it is the kernel of a homomorphism
The "correct way" to think about normal subgroups is as kernels of homomorphisms.

- Patrick Stevens

- Quotient by subgroup is well defined if and only if subgroup is normal
- Order of a group
The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ an…

- Nate Soares - Order of a group element
Given an element $g$ of group $(G, +)$ (which henceforth we abbreviate simply as $G$), the order of …

- Patrick Stevens - Prime order groups are cyclic
This is the first step on the road to classifying the finite groups.

- Patrick Stevens - Simple group
The simple groups form the "building blocks" of group theory, analogously to the prime numbers in number theory.

- Patrick Stevens - Subgroup
A group that lives inside a bigger group.

- Dylan Hendrickson - Symmetric group
The symmetric groups form the fundamental link between group theory and the notion of symmetry.

- Patrick Stevens- Cayley's Theorem on symmetric groups
The "fundamental theorem of symmetry", forging the connection between symmetry and group theory.

- Patrick Stevens - Conjugacy class is cycle type in symmetric group
There is a neat characterisation of the conjugacy classes in the symmetric group on a finite set.

- Patrick Stevens - Conjugacy classes of the symmetric group on five elements
The symmetric group on five elements is a group of just the right size to make a good example of a table of conjugacy classes.

- Patrick Stevens - Cycle notation in symmetric groups
Cycle notation is a convenient way to represent the elements of a symmetric group.

- Patrick Stevens- Cycle type of a permutation
The cycle type is an invariant of a permutation in the symmetric group.

- Patrick Stevens - Disjoint cycle notation is unique
Disjoint cycle notation provides a canonical way to express elements of the symmetric group.

- Patrick Stevens

- Cycle type of a permutation
- Disjoint cycles commute in symmetric groups
In cycle notation, if two cycles are disjoint, then they commute.

- Patrick Stevens - Every member of a symmetric group on finitely many elements is a product of transpositions
This fact can often simplify arguments about permutations: if we can show that something holds for transpositions, and that it holds for products, then it holds for everything.

- Patrick Stevens - Sign homomorphism (from the symmetric group)
The sign homomorphism is how we extract the alternating group from the symmetric group.

- Patrick Stevens - The sign of a permutation is well-defined
This result is what allows the alternating group to exist.

- Patrick Stevens - Transposition (as an element of a symmetric group)
A transposition is the simplest kind of permutation: it swaps two elements.

- Patrick Stevens

- Cayley's Theorem on symmetric groups

- Abelian group
- Group action
"Groups, as men, will be known by their actions."

- Qiaochu Yuan- Group action induces homomorphism to the symmetric group
We can view group actions as "bundles of homomorphisms" which behave in a certain way.

- Patrick Stevens - Group orbits partition
When a group acts on a set, the set falls naturally into distinct pieces, where the group action only permutes elements within any given piece, not between them.

- Patrick Stevens - Orbit-stabiliser theorem
The Orbit-Stabiliser theorem tells us a lot about how a group acts on a given element.

- Patrick Stevens- Orbit-Stabiliser theorem: External Resources
External resources on the Orbit-Stabiliser theorem.

- Mark Chimes

- Orbit-Stabiliser theorem: External Resources
- Stabiliser (of a group action)
If a group acts on a set, it is useful to consider which elements of the group don't move a certain element of the set.

- Patrick Stevens - Stabiliser is a subgroup
Given a group acting on a set, each element of the set induces a subgroup of the group.

- Patrick Stevens

- Group action induces homomorphism to the symmetric group
- Group theory: Examples
What does thinking in terms of group theory actually look like? And what does it buy you?

- Qiaochu Yuan

- Group
- Gödel encoding and self-reference
The formalism that mathematicians use to talk about arithmetic turns out to be able to talk about itself.

- Patrick LaVictoire- Quine
A computer program that prints (or does other computations to) its own source code, using indirect self-reference.

- Patrick LaVictoire

- Quine
- Gödel's first incompleteness theorem
The theorem that destroyed Hilbert's program

- Jaime Sevilla Molina- Proof of Gödel's first incompleteness theorem
##Weak form The weak Gödel's first incompleteness theorem states that every [ $\omega$-consistent] […

- Jaime Sevilla Molina

- Proof of Gödel's first incompleteness theorem
- Iff
If and only if...

- Alexei Andreev - Information theory
The study (and quantificaiton) of information, and its communication and storage.

- Nate Soares- Algorithmic complexity
When you compress the information, what you are left with determines the complexity.

- Eliezer Yudkowsky- Most complex things are not very compressible
We can't *prove* it's impossible, but it would be *extremely surprising* to discover a 500-state Turing machine that output the exact text of "Romeo and Juliet".

- Eliezer Yudkowsky

- Most complex things are not very compressible
- Bit (of data)
A bit of data is the amount of data required to single out one message from a set of two. Equivalen…

- Nate Soares- Bit (of data): Examples
In the game "20 questions", one player (the "leader") thinks of a concept, and the other players ask…

- Nate Soares - Fractional bits
It takes $\log_2(8) = 3$ bits of data to carry one message from a set of 8 possible messages. Simila…

- Nate Soares- Fractional bits: Expected cost interpretation
In the GalCom thought experiment, you regularly have to send large volumes of information through de…

- Nate Soares

- Fractional bits: Expected cost interpretation
- How many bits to a trit?
$\log_2(3) \approx 1.585.$ This can be interpreted a few different ways: 1. If you multiply the nu…

- Nate Soares- Encoding trits with GalCom bits
There are $\log_2(3) \approx 1.585$ bits to a Trit. Why is it that particular value? Consider the Ga…

- Nate Soares

- Encoding trits with GalCom bits

- Bit (of data): Examples
- Communication: magician example
Imagine that you and I are both magicians, performing a trick where I think of a card from a deck of…

- Nate Soares - Compressing multiple messages
How many bits of data does it take to encode an $n$-message? Naively, the answer is $\lceil \log_2(n…

- Nate Soares - Data capacity
The data capacity of an object is defined to be the Logarithm of the number of different distinguish…

- Nate Soares - Decit
Decimal digit

- Nate Soares - Dependent messages can be encoded cheaply
Say you want to transmit a 2-message, a 4-message, and a 256-message to somebody. For example, you m…

- Nate Soares - Fractional bits
It takes $\log_2(8) = 3$ bits of data to carry one message from a set of 8 possible messages. Simila…

- Nate Soares- Fractional bits: Expected cost interpretation
In the GalCom thought experiment, you regularly have to send large volumes of information through de…

- Nate Soares

- Fractional bits: Expected cost interpretation
- GalCom
In the GalCom thought experiment, you live in the future, and make your money by living in the Dene…

- Nate Soares- Encoding trits with GalCom bits
There are $\log_2(3) \approx 1.585$ bits to a Trit. Why is it that particular value? Consider the Ga…

- Nate Soares - GalCom: Rules
1. It costs 1 galcoin per bit to reserve on-peak bits in advance. (Galcoins are very expensive.) 2. …

- Nate Soares

- Encoding trits with GalCom bits
- How many bits to a trit?
$\log_2(3) \approx 1.585.$ This can be interpreted a few different ways: 1. If you multiply the nu…

- Nate Soares- Encoding trits with GalCom bits

- Encoding trits with GalCom bits
- Information
Information is a measure of how much a message grants an observer the ability to predict the world.…

- Nate Soares- Shannon
The shannon (Sh) is a unit of Information. One shannon is the difference in [info\_entropy entropy] …

- Nate Soares

- Shannon
- Intradependent encoding
An encoding $E(m)$ of a message $m$ is intradependent if the fact that $E(m)$ encodes $m$ can be de…

- Nate Soares - Intradependent encodings can be compressed
Given an encoding scheme $E$ which gives an Intradependent encoding of a message $m,$ we can in prin…

- Nate Soares - Trit
Trinary digit

- Nate Soares - n-message
A message singling out one thing from a set of $n$ is sometimes called an $n$-message. For example,…

- Nate Soares

- Algorithmic complexity
- Integer
An **integer** is a Number that can be represented as either a Natural number or its [-additive\_inv…

- Michael Cohen- Integers: Intro (Math 0)
The integers are the whole numbers extended into the negatives.

- Joe Zeng

- Integers: Intro (Math 0)
- Intro to Number Sets
An introduction to number sets for people who have no idea what a number set is.

- Joe Zeng - Irrational number
Real numbers that are not rational numbers

- Joe Zeng- Pi is irrational
The number pi is famously not rational, in spite of joking attempts at legislation to fix its value at 3 or 22/7.

- Patrick Stevens - The square root of 2 is irrational
The number whose square is 2 can't be written is a quotient of natural numbers

- Dylan Hendrickson

- Pi is irrational
- Isomorphism
A morphism between two objects which describes how they are "essentially equivalent" for the purposes of the theory under consideration.

- Mark Chimes- Group isomorphism
"Isomorphism" is the proper notion of "sameness" or "equality" among groups.

- Patrick Stevens - Isomorphism: Intro (Math 0)
Things which are basically the same, except for some stuff you don't care about.

- Mark Chimes - Up to isomorphism
A phrase mathematicians use when saying "we only care about the structure of an object, not about specific implementation details of the object".

- Patrick Stevens

- Group isomorphism
- LaTeX
**Placeholder**

- Eric Bruylant- In notation
There's a weird E-looking symbol called \in in LaTeX. What does it mean?

- Qiaochu Yuan - Mapsto notation
There's an arrow called \mapsto in LaTeX. What does it mean?

- Qiaochu Yuan

- In notation
- Lambda calculus
A minimal, inefficient, and hard-to-read, but still interesting and useful, programming language.

- Dylan Hendrickson - Least common multiple
The **least common multiple (LCM)** of two positive natural numbers a, b is the smallest natural …

- Johannes Schmitt - Linear algebra
The study of [linear\_transformation linear transformations] and vector spaces.

- Nate Soares- Eigenvalues and eigenvectors
We applied this linear transformation to one of its eigenvectors; you won't believe what happened next!

- Zack M. Davis - Subspace
A subspace $U=(F_U, V_U)$ of a Vector space $W=(F_W, V_W)$ is a vector space where $F_U = F_W$ and $…

- Nate Soares - Vector space
A vector space is a field $F$ paired with a Group $V$ and a function $\cdot : F \times V \to V$ (cal…

- Nate Soares- Direct sum of vector spaces
The direct sum of two vector spaces $U$ and $W,$ written $U \oplus W,$ is just the sum of $U$ and $W…

- Nate Soares - Subspace
A subspace $U=(F_U, V_U)$ of a Vector space $W=(F_W, V_W)$ is a vector space where $F_U = F_W$ and $…

- Nate Soares - Sum of vector spaces
The sum of two vector spaces $U$ and $W,$ written $U + W,$ is a vector space where the set of vector…

- Nate Soares

- Direct sum of vector spaces

- Eigenvalues and eigenvectors
- Logarithm
The logarithm base $b$ of a number $n,$ written $\log_b(n),$ is the answer to the question "how man…

- Nate Soares- Exchange rates between digits
In terms of data storage, if a coin is worth $1, a digit wheel is worth more than $3.32, but less than $3.33. Why?

- Nate Soares - Fractional digits
When $b$ and $x$ are integers, $\log_b(x)$ has a few good interpretations. It's roughly the length o…

- Nate Soares - Introductory guide to logarithms
Welcome to the Arbital introduction to logarithms! In modern education, logarithms are often mention…

- Nate Soares - Life in logspace
The log lattice hints at the reason that engineers, scientists, and AI researchers find logarithms s…

- Nate Soares - Log as generalized length
To estimate the log (base 10) of a number, count how many digits it has.

- Nate Soares - Log as the change in the cost of communicating
When interpreting logarithms as a generalization of the notion of "length" and as digit exchange rat…

- Nate Soares - Log base infinity
There is no log base infinity, but if there were, it would send everything to zero

- Nate Soares - Logarithm base 1
There is no log base 1.

- Nate Soares - Logarithm tutorial overview
The logarithm tutorial covers the following six subjects: 1. What are logarithms? 2. Logarithms as…

- Nate Soares - Logarithm: Examples
$\log_{10}(100)=2.$ $\log_2(4)=2.$ $\log_2(3)\approx 1.58.$ (TODO)

- Nate Soares - Logarithm: Exercises
Without using a calculator: What is $\log_{10}(4321)$? What integer is it larger than, what integer …

- Nate Soares - Logarithmic identities
- [ Inversion of exponentials]: $b^{\log_b(n)} = \log_b(b^n) = n.$ - [ Log of 1 is 0]: $\log_b(1) …

- Nate Soares - Logarithms invert exponentials
The function $\log_b(\cdot)$ inverts the function $b^{(\cdot)}.$ In other words, $\log_b(n) = x$ imp…

- Nate Soares - Properties of the logarithm
- $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$ for any $b$, this is the defining characteristic of …

- Nate Soares - The End (of the basic log tutorial)
That concludes our introductory tutorial on logarithms! You have made it to the end. Throughout thi…

- Nate Soares - The characteristic of the logarithm
Any time you find an output that adds whenever the input multiplies, you're probably looking at a (…

- Nate Soares - The log lattice
Log as the change in the cost of communicating and other pages give physical interpretations of what…

- Nate Soares - There is only one logarithm
All logarithm functions are the same, up to a multiplicative constant.

- Nate Soares - What is a logarithm?
Logarithms are a group of functions that take a number as input and produce another number. There i…

- Nate Soares - Why is log like length?
If a number $x$ is $n$ digits long (in Decimal notation), then its logarithm (base 10) is between $n…

- Nate Soares - Why is the decimal expansion of log2(3) infinite?
Because 2 and 3 are relatively prime.

- Nate Soares

- Exchange rates between digits
- Logistic function
A monotonic function from the real numbers to the open unit interval.

- Joe Zeng - Löb's theorem
Löb's theorem

- Jaime Sevilla Molina- Gödel II and Löb's theorem
[ Gödel's second incompleteness theorem] and [ Löb's theorem] are equivalent to each other. ] The …

- Jaime Sevilla Molina - Löb's theorem and computer programs
The close relationship between [ logic and computability] allows us to frame Löb's theorem in terms …

- Jaime Sevilla Molina - Proof of Löb's theorem
Proving that I am Santa Claus

- Jaime Sevilla Molina

- Gödel II and Löb's theorem
- Math 0
Are you not actively bad at math, nor traumatized about math?

- Eliezer Yudkowsky - Math 1
Is math sometimes fun for you, and are you not anxious if you see a math puzzle you don't know how to solve?

- Eliezer Yudkowsky - Math 2
Do you work with math on a fairly routine basis? Do you have little trouble grasping abstract structures and ideas?

- Eliezer Yudkowsky- Math 2 example statements
If you can read these formulas, you're in Math 2!

- Joe Zeng

- Math 2 example statements
- Math 3
Can you read the sort of things that professional mathematicians read, aka LaTeX formulas with a minimum of explanation?

- Eliezer Yudkowsky- Math 3 example statements
If you can read these formulas, you're in Math 3!

- Joe Zeng

- Math 3 example statements
- Math playpen
Playpen page for Math domain

- Alexei Andreev - Mathematical object
**Placeholder**

- Eric Bruylant- Bag
In mathematics, a "bag" is an unordered list. A bag differs from a set in that it can contain the sa…

- Nate Soares - List
A list is an ordered collection of objects, such as `[0, 1, 2, 3]` or `["red", "blue", 0, "shoe"]`. …

- Nate Soares - Number
An abstract object that expresses quantity or value of some sort.

- Joe Zeng- Complex number
A complex number is a number of the form $z = a + b\textrm{i}$, where $\textrm{i}$ is the imaginary …

- Eliana Ruby - Transcendental number
A transcendental number is one which is not the root of any integer-coefficient polynomial.

- Patrick Stevens - Whole number
A term that can refer to three different sets of "numbers that are not fractions".

- Joe Zeng

- Complex number

- Bag
- Metric
A metric is a function that defines a distance between elements in a set and follows some basic rules.

- Bryce Woodworth - Modal logic
The logic of boxes and bots.

- Jaime Sevilla Molina- Kripke model
The semantics of modal logic

- Jaime Sevilla Molina - Modal combat
Modal combat

- Jaime Sevilla Molina - Modalized modal sentence
A [ modal sentence] $A$ is said to be **modalized** in $p$ if every occurrence of $p$ happens within…

- Jaime Sevilla Molina - Normal system of provability logic
Between the modal systems of provability, the normal systems distinguish themselves by exhibiting ni…

- Jaime Sevilla Molina - Provability logic
Learn how to reason about provability!

- Jaime Sevilla Molina- Fixed point theorem of provability logic
Deal with those pesky self-referential sentences!

- Jaime Sevilla Molina - Solovay's theorems of arithmetical adequacy for GL
Using GL to reason about PA, and viceversa

- Jaime Sevilla Molina

- Fixed point theorem of provability logic
- Standard provability predicate
Encoding provability as a statement of arithmetic

- Jaime Sevilla Molina

- Kripke model
- Modular arithmetic
Addition as traveling around a circle, instead of along a line.

- Malcolm McCrimmon - Morphism
A morphism is the abstract representation of a relation between mathematical objects. Usually, it i…

- Jaime Sevilla Molina - Multiplication of rational numbers (Math 0)
"Multiplication" is the idea of "now do the same as you just did, but instead of doing it to one apple, do it to some other number".

- Patrick Stevens - Natural number
The numbers we use to count: 0, 1, 2, 3, ...

- Jaime Sevilla Molina- A googol
A pretty small large number.

- Nate Soares - A googolplex
A moderately large number, as large numbers go.

- Nate Soares - Graham's number
A fairly large number, as numbers go.

- Nate Soares - Prime number
The prime numbers are the "building blocks" of the counting numbers.

- Patrick Stevens- Proof that there are infinitely many primes
Suppose there were finitely many primes. Then consider the product of all the primes plus 1...

- Joe Zeng

- Proof that there are infinitely many primes

- A googol
- Order of operations
Conventions used for disambiguating infix notation.

- Joe Zeng- Order of rational operations (Math 0)
Our shorthand for all the operations on rationals is very useful, but full of brackets; this is how to get rid of some of the brackets.

- Patrick Stevens

- Order of rational operations (Math 0)
- Order theory
The study of binary relations that are reflexive, transitive, and antisymmetic.

- Kevin Clancy- Complete lattice
A poset that is closed under arbitrary joins and meets.

- Kevin Clancy - Join and meet
Let $\langle P, \leq \rangle$ be a poset, and let $S \subseteq P$. The **join** of $S$ in $P$, deno…

- Kevin Clancy- Join and meet: Examples
A union of sets and the least common multiple of a set of natural numbers can both be viewed as join…

- Kevin Clancy - Join and meet: Exercises
Try these exercises to test your knowledge of joins and meets. Tangled up -------------------- !…

- Kevin Clancy

- Join and meet: Examples
- Lattice (Order Theory)
A poset that is closed under binary joins and meets.

- Kevin Clancy- Lattice: Examples
Here are some additional examples of lattices. $\newcommand{\nsubg}{\mathcal N \mbox{-} Sub~G}$ A f…

- Kevin Clancy - Lattice: Exercises
Try these exercises to test your knowledge of lattices. ## Distributivity Does the lattice meet op…

- Kevin Clancy

- Lattice: Examples
- Monotone function
An order-preserving map between posets.

- Kevin Clancy- Monotone function: examples
Here are some examples of monotone functions. A cunning plan -------- There's a two-player game ca…

- Kevin Clancy - Monotone function: exercises
Try these exercises and become a *deity* of monotonicity. Monotone composition ----- Let $P, Q$, …

- Kevin Clancy

- Monotone function: examples
- Partially ordered set
A set endowed with a relation that is reflexive, transitive, and antisymmetric.

- Kevin Clancy- Greatest lower bound in a poset
The greatest lower bound is an abstraction of the idea of the greatest common divisor to a general poset.

- Patrick Stevens - Poset: Examples
The standard $\leq$ relation on integers, the $\subseteq$ relation on sets, and the $|$ (divisibilit…

- Kevin Clancy - Poset: Exercises
Try these exercises to test your poset knowledge. # Corporate Ladder Imagine a company with five …

- Kevin Clancy

- Greatest lower bound in a poset
- Totally ordered set
A set where all the elements can be compared as greater than or less than.

- Joe Zeng- Well-ordered set
An ordered set with an order that always has a "next element".

- Dylan Hendrickson

- Well-ordered set

- Complete lattice
- Ordered field
An ordered ring with division.

- Joe Zeng - Ordering of rational numbers (Math 0)
How do we know if one lot of apples is "more apples" than another lot?

- Patrick Stevens - P vs NP
Is creativity purely mechanical?

- Jaime Sevilla Molina- P vs NP: Arguments against P=NP
Why we believe P and NP are different

- Jaime Sevilla Molina

- P vs NP: Arguments against P=NP
- Pi
Pi, usually written $π$, is a number equal to the ratio of a circle's [-circumference] to its [-diam…

- Michael Cohen- Pi is irrational
The number pi is famously not rational, in spite of joking attempts at legislation to fix its value at 3 or 22/7.

- Patrick Stevens

- Pi is irrational
- Primer on Infinite Series
What does it mean to add things together forever?

- Chris Holden - Probability theory
The logic of science; coherence relations on quantitative degrees of belief.

- Eliezer Yudkowsky- Bayesian reasoning
A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.

- Eliezer Yudkowsky- Bayes' rule
Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.

- Eliezer Yudkowsky- Bayes' rule examples
Interesting problems solvable by Bayes' rule

- Eliezer Yudkowsky- Introductory Bayesian problems
Bayesian problems to try to solve yourself, before beginning to learn about Bayes' rule.

- Eliezer Yudkowsky- Blue oysters
A probability problem about blue oysters.

- Nate Soares - Diseasitis
20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What's the chance you have Diseasitis?

- Eliezer Yudkowsky - Sock-dresser search
There's a 4/5 chance your socks are in one of your dresser's 8 drawers. You check 6 drawers at random. What's the probability they'll be in the next drawer you check?

- Nate Soares - Sparking widgets
10% of widgets are bad and 90% are good. 4% of good widgets emit sparks, and 12% of bad widgets emit…

- Nate Soares

- Blue oysters
- Realistic (Math 1)
Real-life examples of Bayesian reasoning

- Eliezer Yudkowsky

- Introductory Bayesian problems
- Bayes' rule: Beginner's guide
Beginner's guide to learning about Bayes' rule.

- Alexei Andreev - Bayes' rule: Definition
Bayes' rule is the mathematics of probability theory governing how to update your beliefs in the lig…

- Nate Soares - Bayes' rule: Functional form
Bayes' rule for to continuous variables.

- Eliezer Yudkowsky - Bayes' rule: Guide
The Arbital guide to Bayes' rule

- Eliezer Yudkowsky- Bayes' Rule and its different forms
This is an arc that includes different ways to look at Bayes' Rule.

- Alexei Andreev - Bayes' Rule and its implications
This is an arc that includes implications of the Bayes's Rule.

- Alexei Andreev - Comprehensive guide to Bayes' Rule
This is an arc that includes all Bayes content.

- Alexei Andreev - Introduction to Bayes' Rule odds form
This is an arc that includes just enough content to teach about Bayes's Rule odds form.

- Alexei Andreev - Wants to get straight to Bayes
A simple requisite page to mark whether the user has selected wanting to get straight into Bayes on …

- Eliezer Yudkowsky

- Bayes' Rule and its different forms
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.

- Eliezer Yudkowsky - Bayes' rule: Odds form
The simplest and most easily understandable form of Bayes' rule uses relative odds.

- Eliezer Yudkowsky- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.

- Eliezer Yudkowsky

- Introduction to Bayes' rule: Odds form
- Bayes' rule: Probability form
The original formulation of Bayes' rule.

- Nate Soares- Odds form to probability form
The odds form of Bayes' rule works for any two hypotheses $H_i$ and $H_j,$ and looks like this: $$\…

- Nate Soares - Proof of Bayes' rule: Probability form
Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_…

- Nate Soares

- Odds form to probability form
- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.

- Eliezer Yudkowsky - Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)

- Eliezer Yudkowsky - Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.

- Eliezer Yudkowsky - Frequency diagram
Visualizing Bayes' rule by manipulating frequencies in large populations

- Nate Soares- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.

- Nate Soares

- Frequency diagrams: A first look at Bayes
- High-speed intro to Bayes's rule
A high-speed introduction to Bayes's Rule on one page, for the impatient and mathematically adept.

- Eliezer Yudkowsky - Path: Multiple angles on Bayes's Rule
A learning-path placeholder page for learning multiple angles on Bayes's Rule.

- Eliezer Yudkowsky - Probability notation for Bayes' rule
The probability notation used in Bayesian reasoning

- Eliezer Yudkowsky- Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.

- Eliezer Yudkowsky

- Probability notation for Bayes' rule: Intro (Math 1)
- Proof of Bayes' rule
Proofs of Bayes' rule, with graphics

- Eliezer Yudkowsky- Proof of Bayes' rule: Probability form
Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_…

- Nate Soares

- Proof of Bayes' rule: Probability form
- Shift towards the hypothesis of least surprise
When you see new evidence, ask: which hypothesis is *least surprised?*

- Nate Soares - Waterfall diagram
Visualizing Bayes' rule as the mixing of probability streams.

- Eliezer Yudkowsky- Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems

- Eliezer Yudkowsky

- Waterfall diagrams and relative odds

- Bayes' rule examples
- Bayesian update
Bayesian updating: the ideal way to change probabilistic beliefs based on evidence.

- Eliezer Yudkowsky- Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.

- Eliezer Yudkowsky - Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.

- Eliezer Yudkowsky- Extraordinary claims
What makes something an 'extraordinary claim' that requires extraordinary evidence?

- Eliezer Yudkowsky

- Extraordinary claims
- Ordinary claims require ordinary evidence
Extraordinary claims require extraordinary evidence, but ordinary claims *don't*.

- Nate Soares - Path: Insights from Bayesian updating
A learning-path placeholder page for insights derived from the Bayesian rule for updating beliefs.

- Eliezer Yudkowsky - Strength of Bayesian evidence
From a Bayesian standpoint, the strength of evidence can be identified with its likelihood ratio.

- Eliezer Yudkowsky

- Bayesian view of scientific virtues
- Empirical probabilities are not exactly 0 or 1
"Cromwell's Rule" says that probabilities of exactly 0 or 1 should never be applied to empirical propositions - there's always some probability, however tiny, of being mistaken.

- Eliezer Yudkowsky - Finishing your Bayesian path on Arbital
The page that comes at the end of reading the Arbital Guide to Bayes' rule

- Eliezer Yudkowsky - Humans doing Bayes
The human use of Bayesian reasoning in everyday life

- Eliezer Yudkowsky- Explicit Bayes as a counter for 'worrying'
Explicitly walking through Bayes's Rule can summarize your knowledge and thereby stop you from bouncing around pieces of it.

- Eliezer Yudkowsky

- Explicit Bayes as a counter for 'worrying'
- Ignorance prior
Key equations for quantitative Bayesian problems, describing exactly the right shape for what we believed before observation.

- Eliezer Yudkowsky- Inductive prior
Some states of pre-observation belief can learn quickly; others never learn anything. An "inductive prior" is of the former type.

- Eliezer Yudkowsky- Laplace's Rule of Succession
Suppose you flip a coin with an unknown bias 30 times, and see 4 heads and 26 tails. The Rule of Succession says the next flip has a 5/32 chance of showing heads.

- Eliezer Yudkowsky - Solomonoff induction
A simple way to superintelligently predict sequences of data, given unlimited computing power.

- Eliezer Yudkowsky- Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math

- Eliezer Yudkowsky

- Solomonoff induction: Intro Dialogue (Math 2)
- Universal prior
A "universal prior" is a probability distribution containing *all* the hypotheses, for some reasonable meaning of "all". E.g., "every possible computer program that computes probabilities".

- Eliezer Yudkowsky

- Laplace's Rule of Succession

- Inductive prior
- Interest in mathematical foundations in Bayesianism
"Want" this requisite if you prefer to see extra information about the mathematical foundations in Bayesianism.

- Eliezer Yudkowsky - Likelihood
"Likelihood", when speaking of Bayesian reasoning, denotes *the probability of an observation, sup…

- Nate Soares- Likelihood function
Let's say you have a piece of evidence $e$ and a set of hypotheses $\mathcal H.$ Each $H_i \in \math…

- Nate Soares - Likelihood notation
The likelihood of a piece of evidence $e$ according to a hypothesis $H,$ known as "the likelihood of…

- Nate Soares - Likelihood ratio
Given a piece of evidence $e$ and two hypothsese $H_i$ and $H_j,$ the likelihood ratio between them…

- Nate Soares - Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.

- Eliezer Yudkowsky

- Likelihood function
- Posterior probability
What we believe, after seeing the evidence and doing a Bayesian update.

- Eliezer Yudkowsky - Prior
A state of prior knowledge, before seeing information on a new problem. Potentially complicated.

- Eliezer Yudkowsky - Prior probability
What we believed before seeing the evidence.

- Eliezer Yudkowsky - Strictly confused
A hypothesis is strictly confused by the raw data, if the hypothesis did much worse in predicting it than the hypothesis itself expected.

- Eliezer Yudkowsky - Subjective probability
Probability is in the mind, not in the environment. If you don't know whether a coin came up heads or tails, that's a fact about you, not a fact about the coin.

- Eliezer Yudkowsky- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Nate Soares

- Probability distribution: Motivated definition

- Bayes' rule
- Mutually exclusive and exhaustive
The condition needed for probabilities to sum to 1

- Eliezer Yudkowsky - Normalization (probability)
That thingy we do to make sure our probabilities sum to 1, when they should sum to 1.

- Eliezer Yudkowsky - Odds
Odds express a relative probability.

- Eliezer Yudkowsky- Odds: Introduction
What's the difference between probabilities and odds? Why is a 20% probability of success equivalent to 1 : 4 odds favoring success?

- Nate Soares - Odds: Refresher
A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).

- Nate Soares - Odds: Technical explanation
Formal definitions, alternate representations, and uses of odds and odds ratios (like a 1 : 2 chance of drawing a red ball vs. green ball from a barrel).

- Alexei Andreev

- Odds: Introduction
- Probability
The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.

- Eliezer Yudkowsky- Conditional probability
The notation for writing "The probability that someone has green eyes, if we know that they have red hair."

- Eliezer Yudkowsky- Conditional probability: Refresher
Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?

- Nate Soares

- Conditional probability: Refresher
- Interpretations of "probability"
What does it *mean* to say that a fair coin has a 50% probability of coming up heads?

- Nate Soares- Correspondence visualizations for different interpretations of "probability"
Let's say you have a model which says a particular coin is 70% likely to be heads. How should we as…

- Nate Soares - Probability interpretations: Examples
Consider evaluating, in June of 2016, the question: "What is the probability of Hillary Clinton wi…

- Nate Soares - Subjective probability
Probability is in the mind, not in the environment. If you don't know whether a coin came up heads or tails, that's a fact about you, not a fact about the coin.

- Eliezer Yudkowsky- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Nate Soares

- Probability distribution: Motivated definition

- Correspondence visualizations for different interpretations of "probability"
- Joint probability
The notation for writing the chance that both X and Y are true.

- Eliezer Yudkowsky - Report likelihoods, not p-values
If scientists reported likelihood functions instead of p-values, this could help science avoid p-ha…

- Nate Soares- Likelihood functions, p-values, and the replication crisis
What's the whole Bayesian-vs.-frequentist debate about?

- Eliezer Yudkowsky - Report likelihoods not p-values: FAQ
This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…

- Nate Soares

- Likelihood functions, p-values, and the replication crisis

- Conditional probability
- Probability distribution (countable sample space)
A function assigning a probability to each point in the sample space.

- Tsvi BT - Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Nate Soares - Sample space
The set of possible things that could happen in a part of the world that you are uncertain about.

- Tsvi BT- Sample spaces are too large
Sample spaces are often large, so it is hard to do probabilistic computations using a raw distribution over the sample space.

- Tsvi BT - Uncountable sample spaces are way too large
We can't define probability distributions over uncountable sample spaces by just assigning numbers to each point in the sample space.

- Tsvi BT

- Sample spaces are too large
- Square visualization of probabilities on two events
$$ \newcommand{\true}{\text{True}} \newcommand{\false}{\text{False}} \newcommand{\bP}{\mathbb{P}} …

- Tsvi BT- Square visualization of probabilities on two events: (example) Diseasitis
But it *seems* like the patient with the black tongue depressor has diseasitis...

- Tsvi BT

- Square visualization of probabilities on two events: (example) Diseasitis
- Two independent events
What do [a pair of dice], [a pair of coins], and [a pair of people on opposite sides of the planet] all have in common?

- Tsvi BT- Two independent events: Square visualization
$$ \newcommand{\true}{\text{True}} \newcommand{\false}{\text{False}} \newcommand{\bP}{\mathbb{P}} …

- Tsvi BT

- Two independent events: Square visualization
- Well-calibrated probabilities
Even if you're fairly ignorant, you can still strive to ensure that when you say "70% probability", it's true 70% of the time.

- Eliezer Yudkowsky

- Bayesian reasoning
- Proof technique
**Placeholder**

- Eric Bruylant- Mathematical induction
Proving a statement about all positive integers by knocking them down like dominoes.

- Douglas Weathers - Proof by contradiction
Discover what 'reductio ad absurdum' means!

- Jaime Sevilla Molina

- Mathematical induction
- Proportion
A representation of a value as a fraction or multiple of another value.

- Joe Zeng - Provability predicate
A provability predicate of a theory $T$ is a formula $P(x)$ with one free variable $x$ such that: …

- Jaime Sevilla Molina - Quotient group
Given a group $G$ with operation $\bullet$ and a special kind of subgroup $N \leq G$ called the "no…

- Adele Lopez - Rational number
The rational numbers are "fractions".

- Patrick Stevens- Arithmetic of rational numbers (Math 0)
How do we combine rational numbers together?

- Patrick Stevens- Addition of rational numbers (Math 0)
The simplest operation on rational numbers is addition.

- Patrick Stevens- Addition of rational numbers exercises
Test and cement your understanding of how we add rational numbers!

- Patrick Stevens

- Addition of rational numbers exercises
- Division of rational numbers (Math 0)
"Division" is the idea of "dividing something up among some people so that we can give equal amounts to each person".

- Patrick Stevens - Order of rational operations (Math 0)
Our shorthand for all the operations on rationals is very useful, but full of brackets; this is how to get rid of some of the brackets.

- Patrick Stevens - Rational arithmetic all works together
The various operations of arithmetic all play nicely together in a certain specific way.

- Patrick Stevens - Subtraction of rational numbers (Math 0)
In which we meet anti-apples.

- Patrick Stevens

- Addition of rational numbers (Math 0)
- Field structure of rational numbers
In which we describe the field structure on the rationals.

- Patrick Stevens- Arithmetic of rational numbers (Math 0)
How do we combine rational numbers together?

- Patrick Stevens- Addition of rational numbers (Math 0)
The simplest operation on rational numbers is addition.

- Patrick Stevens- Addition of rational numbers exercises
Test and cement your understanding of how we add rational numbers!

- Patrick Stevens

- Addition of rational numbers exercises
- Division of rational numbers (Math 0)
"Division" is the idea of "dividing something up among some people so that we can give equal amounts to each person".

- Patrick Stevens - Order of rational operations (Math 0)
Our shorthand for all the operations on rationals is very useful, but full of brackets; this is how to get rid of some of the brackets.

- Patrick Stevens - Rational arithmetic all works together
The various operations of arithmetic all play nicely together in a certain specific way.

- Patrick Stevens - Subtraction of rational numbers (Math 0)
In which we meet anti-apples.

- Patrick Stevens

- Addition of rational numbers (Math 0)

- Arithmetic of rational numbers (Math 0)
- Rational numbers: Intro (Math 0)
The rational numbers are "fractions". While the natural numbers measure the answer to the question …

- Patrick Stevens - The rationals form a field
The set $\mathbb{Q}$ of rational numbers is a field. # Proof $\mathbb{Q}$ is a (commutative) ring …

- Patrick Stevens

- Arithmetic of rational numbers (Math 0)
- Real analysis
The study of real numbers and real-valued functions.

- Kevin Clancy - Real number
A **real number** is any number that can be used to represent a physical quantity. Intuitively, rea…

- Michael Cohen- Real number (as Cauchy sequence)
There are several ways to construct real numbers; this is the most natural way to use them in computations.

- Patrick Stevens- The reals (constructed as classes of Cauchy sequences of rationals) form a field
The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.

- Patrick Stevens

- The reals (constructed as classes of Cauchy sequences of rationals) form a field
- Real number (as Dedekind cut)
A way to construct the real numbers that follows the intuition of filling in the gaps.

- Joe Zeng- The reals (constructed as Dedekind cuts) form a field
The reals are an archetypal example of a field, but if we are to construct them from simpler objects, we need to show that our construction does indeed have the right properties.

- Patrick Stevens

- The reals (constructed as Dedekind cuts) form a field
- Real numbers are uncountable
The real numbers are uncountable.

- Eric Bruylant

- Real number (as Cauchy sequence)
- Relation
A **relation** is a set of [tuple\_mathematics tuples], all of which have the same [tuple\_arity ar…

- Kevin Clancy- Antisymmetric relation
A binary relation where no two distinct elements are related in both directions

- M Yass - Equivalence relation
A relation that allows you to partition a set into equivalence classes.

- Dylan Hendrickson - Order relation
A way of determining which elements of a set come "before" or "after" other elements.

- Joe Zeng - Reflexive relation
A binary relation over some set is **reflexive** when every element of that set is related to itself…

- Ryan Hendrickson - Transitive relation
If a is related to b and b is related to c, then a is related to c.

- Dylan Hendrickson

- Antisymmetric relation
- Set
An unordered collection of distinct objects.

- Nate Soares- Cantor-Schröder-Bernstein theorem
This theorem tells us that comparing sizes of sets makes sense: if one set is smaller than another, and the other is smaller than the one, then they are the same size.

- Patrick Stevens - Cardinality
The "size" of a set, or the "number of elements" that it has.

- Joe Zeng - Convex set
A set that contains all line segments between points in the set

- Jessica Taylor - Disjoint union of sets
One of the most basic ways we have of joining two sets together.

- Patrick Stevens- Universal property of the disjoint union
Just as the empty set may be described by a universal property, so too may the disjoint union of sets.

- Patrick Stevens

- Universal property of the disjoint union
- Empty set
The empty set does what it says on the tin: it is the set which is empty.

- Patrick Stevens- Empty set
The empty set, $\emptyset$, is the set with no elements. For every object $x$, $x$ is not in $\empt…

- Patrick Stevens

- Empty set
- Extensionality Axiom
If two sets have exactly the same members, then they are equal

- Ilia Zaichuk - Finite set
A finite set is one which is not infinite. Some of these are the least complicated sets.

- Patrick Stevens- Category of finite sets
The category of finite sets is exactly what it claims to be. It's a useful training ground for some of the ideas of category theory.

- Patrick Stevens

- Category of finite sets
- Operations in Set theory
An operation in set theory is a Function of two sets, that returns a set. Common set operations inc…

- M Yass- Absolute Complement
The complement $A^\complement$ of a set $A$ is the set of all things that are not in $A$. Put simply…

- M Yass - Cartesian product
The Cartesian product of two sets $A$ and $B,$ denoted $A \times B,$ is the set of all [ordered\_pai…

- Nate Soares - Intersection
The intersection of two sets is the set of elements they have in common

- M Yass - Relative complement
The relative complement of two sets $A$ and $B$, denoted $A \setminus B$, is the set of elements tha…

- M Yass - Union
The union of two sets is the set of elements which are in one or the other, or both

- M Yass

- Absolute Complement
- Set builder notation
$\{ 2n \mid n \in \mathbb N \}$ denotes the set of all even numbers, using set builder notation. Set…

- Nate Soares - Set product
A fundamental way of combining sets is to take their product, making a set that contains all tuples of elements from the originals.

- Patrick Stevens

- Cantor-Schröder-Bernstein theorem
- Square root
What is the opposite of multiplying a number by itself?

- Travis Rivera - String (of text)
A string (of text) is a series of letters (often denoted by quote marks), such as `"abcd"` or `"hell…

- Nate Soares - The n-th root of m is either an integer or irrational
In other words, no power of a rational number that is not an integer is ever an integer.

- Joe Zeng - The set of rational numbers is countable
Although there are "lots and lots" of rational numbers, there are still only countably many of them.

- Patrick Stevens - Trit
Trinary digit

- Nate Soares - Turing machine
A Turing Machine is a simple mathematical model of computation that is powerful enough to describe any computation a computer can do.

- Eric Leese- Rice's Theorem
Rice's Theorem tells us that if we want to determine pretty much anything about the behaviour of an arbitrary computer program, we can't in general do better than just running it.

- Patrick Stevens- Proof of Rice's theorem
A standalone proof of Rice's theorem, including one surprising lemma.

- Patrick Stevens - Rice's Theorem: Intro (Math 1)
You can't write a program that looks at another programs source code, and tells you whether it computes the Fibonacci sequence.

- Dylan Hendrickson - Rice's theorem and the Halting problem
We will show that Rice's theorem and the the halting problem are equivalent. #The Halting theorem i…

- Jaime Sevilla Molina

- Proof of Rice's theorem
- Turing machine: External resources
* [Wikipedia](https://en.wikipedia.org/wiki/Turing_machine) * [Wolfram MathWorld](http://mathworld.w…

- Eric Bruylant

- Rice's Theorem
- Type theory
Modern foundations for formal mathematics.

- Jack Gallagher- Programming in Dependent Type Theory
Working with simple types in Lean

- Jack Gallagher

- Programming in Dependent Type Theory
- Uncomputability
The diagonal function and the halting problem

- Jaime Sevilla Molina - Uncountability
Some infinities are bigger than others. Uncountable infinities are larger than countable infinities.

- Jason Gross- Real numbers are uncountable
The real numbers are uncountable.

- Eric Bruylant - Uncountability (Math 3)
Formal definition of uncountability, and foundational considerations.

- Patrick Stevens - Uncountability: Intro (Math 1)
Not all infinities are created equal. The infinity of real numbers is infinitely larger than the infinity of counting numbers.

- Jason Gross - Uncountability: Intuitive Intro
Are all sizes of infinity the same? What does "the same" even mean here?

- Jason Gross

- Real numbers are uncountable
- Well-defined
A mathematical object is "well-defined" if we have given it a completely unambiguous definition.

- Patrick Stevens - n-digit
An $n$-digit is a physical object that can be stably placed into any of $n$ distinguishable states. …

- Nate Soares

## Mid Term

- child
test

- Alex Foster - this is a claim - Alex Foster

## Octalysis Framework

- If you want to make a task fun you will need to craft and combine different feelings with care. - Sarah Wolf
- Octalysis Framework is useful for identifying what game mechanics will lead to different feelings - Sarah Wolf
- The Octalysis Framework will not tell you which combinations of feelings will lead to good gameplay. - Sarah Wolf
- game mechanics are tools that manipulate the players feelings rather than tools to make a task fun. - Sarah Wolf

## Omniment

- Hiring
What's Omniment's protocol for hiring employees?

- Alexei Andreev

## On the importance of Less Wrong, or another single conversational locus

- In order to have a shared conversation, it would help to have a Schelling set of posts. - Eric Rogstad
- In order to think accumulatively we must have a shared conversation. - Eric Rogstad
- Our community has a shot at seriously helping with x-risk. - Eric Rogstad
- The world needs something like a miracle of good thought if it is to have decent odds of overcoming x-risk. - Eric Rogstad
- Thinking accumulatively is one of the keys to solving existential risks. - Eric Rogstad
- We do not currently have a single shared conversation. - Eric Rogstad

## Our community should relocate to a country other than the US

- It's easier to make scientific progress in countries with fewer regulations. - Alexei Andreev
- Moving outside of the US will likely result in cheaper accommodations and living costs. - Alexei Andreev
- Our community should relocate to Iceland. - Alexei Andreev
- Our community should relocate to Japan. - Alexei Andreev
- The benefits of moving the community won't be realized in time before AGI. - Alexei Andreev
- The community would lose a lot of its influence if it doesn't have a strong presence in key geographical locations (e.g. Bay Area). - Alexei Andreev
- To make faster intellectual progress, it would help if the community was located more densely together. - Alexei Andreev
- To think thoughts fully uninfluenced by the surrounding society one has to be physically removed from it.
This claim is related to Our community should relocate to a country other than the US.

- Alexei Andreev

## Paul Christiano's AI control blog

- A possible stance for AI control research
I think that AI control research should focus on finding [scalable](https://arbital.com/pages/492374…

- Paul Christiano - Abstract approval-direction
Consider the following design for an agent, which I first described [here](https://arbital.com/p/1t7…

- Paul Christiano- Learning representations
Many AI systems form internal representations of their current environment or of particular data. Pr…

- Paul Christiano

- Learning representations
- Act based agents
I’ve recently discussed three kinds of learning systems: - [Approval-directed agents](https://arbit…

- Paul Christiano- Imitation-based agent
An AI meant to imitate the behavior of a reference human as closely as possible.

- Eliezer Yudkowsky

- Imitation-based agent
- Adversarial collaboration
Suppose that I have hired a group of employees who are much smarter than I am. For some tasks it’s…

- Paul Christiano - Against mimicry
One simple and apparently safe AI system is a “copycat:” an agent that predicts what its user woul…

- Paul Christiano - Apprenticeship learning and mimicry
This post compares my [recent proposal](https://arbital.com/p/1vp/mimicry_meeting_halfway) with [Ab…

- Paul Christiano - Approval directed agents
Research in AI is steadily progressing towards more flexible, powerful, and autonomous goal-directe…

- Paul Christiano - Approval-directed bootstrapping
Approval-directed behavior works best when the overseer is very smart. Where can we find a smart o…

- Paul Christiano - Automated assistants
In my [last post](https://arbital.com/p/1th?title=implementing-our-considered-judgment), I describ…

- Paul Christiano - Challenges for safe AI from RL
In this post, I’ll describe and discuss a few big problems for the proposal from [my last post](htt…

- Paul Christiano - Delegating to a mixed crowd
### Suppose I have ten programs, each a human-level agent. I suspect that at least one or two of…

- Paul Christiano - Efficient feedback
In some machine learning domains, such as image classification, we can produce a bunch of labelled t…

- Paul Christiano - Handling adversarial errors
Even a very powerful learning system can’t do everything perfectly at first — it requires time to l…

- Paul Christiano- Handling errors with arguments
My [recent proposal](https://arbital.com/p/1v7?title=steps-towards-safe-ai-from-online-learning) f…

- Paul Christiano

- Handling errors with arguments
- How common is imitation?
How often do we train machine learning systems to imitate human behavior? Some researchers explicit…

- Paul Christiano - Human arguments and AI control
### Explanation and AI control Consider the definition: > An action is good to the extent that I w…

- Paul Christiano - Human in counterfactual loop
Consider an autonomous system which is buying or selling assets, operating heavy machinery, or mak…

- Paul Christiano - Humans consulting HCH
Consider a human who has access to a question-answering machine. Suppose the machine answers questio…

- Paul Christiano - Implementing our considered judgment
Suppose I had a very powerful prediction algorithm. How might I use this algorithm to build a smar…

- Paul Christiano - Implicit consequentialism
Consider a machine that does exactly what its user [would tell it to do](https://arbital.com/p/1tj?t…

- Paul Christiano - In defense of maximization
I’ve been thinking about [AI systems that take actions their users would most approve of](https://…

- Paul Christiano - Indirect decision theory
In which I argue that understanding decision theory can be delegated to AI. ### Indirect normativit…

- Paul Christiano - Learn policies or goals?
I’ve [recently proposed](https://arbital.com/p/1t7/approval_directed_agents) training agents to ma…

- Paul Christiano - Learning and logic
In most machine learning tasks, the learner maximizes a concrete, empirical performance measure: i…

- Paul Christiano - Mimicry and meeting halfway
I’ve talked recently about two different model-free decision procedures: - At each step, pick the …

- Paul Christiano - Modeling AI control with humans
I’ve been trying to build an aligned AI out of reward-maximizing modules. A successful scheme could …

- Paul Christiano - Of simulations and inductive definitions
_(Warning: weird.)_ Consider a simple AI system, named A, that carries out a task by predicting wha…

- Paul Christiano - On heterogeneous objectives
Eliezer Yudkowsky [has said](https://www.facebook.com/yudkowsky/posts/10153748345169228): > If you …

- Paul Christiano - Optimization and goals
If we want to write a program that _doesn’t_ pursue a goal, we can have two kinds of trouble: 1. We…

- Paul Christiano - Optimizing with comparisons
I could [elicit a user’s approval](https://arbital.com/p/1w5) of an action _a_ by having them supply…

- Paul Christiano - Problem: safe AI from episodic RL
In [a previous post](https://arbital.com/p/1tv?title=the-steering-problem), I posed the steering pr…

- Paul Christiano- Steps towards safe AI from online learning
### Steps towards safe AI from online learning Suppose that we have a good algorithm for episodic r…

- Paul Christiano

- Steps towards safe AI from online learning
- Reinforcement learning and linguistic convention
Existing machine learning techniques are most effective when we can provide concrete feedback — such…

- Paul Christiano - Reward engineering
This post gestures at a handful of research questions with a loose thematic connection. ### The id…

- Paul Christiano- Technical and social approaches to AI safety
I often divide solutions to the AI control problem into two parts: technical and social. I think a…

- Paul Christiano

- Technical and social approaches to AI safety
- Safe AI from question-answering
_(Warning: minimal new content. Just a clearer framing.)_ Suppose that I have a question-answering…

- Paul Christiano - Scalable AI control
By AI control, I mean the problem of getting AI systems to do what we want them to do, to the best…

- Paul Christiano - Stable self-improvement as an AI safety problem
“Stable self-improvement” seems to be a primary focus of MIRI’s work. As I understand it, the proble…

- Paul Christiano - Synthesizing training data
[Counterfactual oversight](https://arbital.com/p/1tj?title=human-in-counterfactual-loop) requires th…

- Paul Christiano - The absentee billionaire
Once each day, Hugh wakes for 10 minutes. During these 10 minutes, he spends 10 million dollars. The…

- Paul Christiano - The easy goal inference problem is still hard
Goal inference and inverse reinforcement learning ------------------------------------------------…

- Paul Christiano- Online guarantees and AI control
I’m interested in claims of the form: “If we had an AI that could do X well, then we could build a…

- Paul Christiano

- Online guarantees and AI control
- The state of the steering problem
The [steering problem](https://arbital.com/p/1tv?title=the-steering-problem) asks: given some powe…

- Paul Christiano - The steering problem
Most AI research focuses on reproducing human abilities: to learn, infer, and reason; to perceive,…

- Paul Christiano - Unsupervised learning and AI control
Reinforcement learning systems optimize for an objective defined by external feedback — anything fro…

- Paul Christiano

## People

- Nick Bostrom
Nick Bostrom, secretly the inventor of Friendly AI

- Eliezer Yudkowsky

## Philosophy

- Descriptive versus normative propositions
A normative proposition talks about what should be; a descriptive proposition talks about what is.

- Eliezer Yudkowsky - Intution pump
In philosophy, a metaphor or visualization used to shove the listener's intuition in a particular direction.

- Eliezer Yudkowsky - Metaethics
Metaethics asks "What kind of stuff is goodness made of?" (or "How would we compute goodness?") rather than "Which particular policies or outcomes are good or not-good?"

- Eliezer Yudkowsky- Terminal versus instrumental goals / values / preferences
Distinguish events wanted for their consequences, from events wanted locally.

- Eliezer Yudkowsky

- Terminal versus instrumental goals / values / preferences

## Predictions For 2017

- 2017 US GDP growth will be lower than in 2016 - Alexei Andreev
- 2017 will have no interesting progress with Gaza or peace negotiations in general - Alexei Andreev
- 2017 will not have any major revolt (greater than or equal to Tiananmen Square) against Chinese Communist Party - Alexei Andreev
- Angela Merkel will be re-elected Chancellor of Germany in 2017 - Alexei Andreev
- Bitcoin will end 2017 higher than $1000 - Alexei Andreev
- By end of 2017 ISIS will control less territory than it did at the beginning of the year - Alexei Andreev
- Donald Trump remains President at the end of 2017 - Alexei Andreev
- Donald Trump’s approval rating at the end of 2017 is lower than fifty percent - Alexei Andreev
- Donald Trump’s approval rating at the end of 2017 is lower than forty percent - Alexei Andreev
- Dow Jones will not end 2017 down by more than 10% - Alexei Andreev
- Fewer refugees admitted by Europe in 2017 than 2016 - Alexei Andreev
- France will not declare a plan to leave EU in 2017 - Alexei Andreev
- Germany will not declare a plan to leave EU - Alexei Andreev
- ISIS will not continue to exist as a state entity in Iraq/Syria by end of 2017 - Alexei Andreev
- In 2017 SSC will get fewer hits than in 2016 - Alexei Andreev
- In 2017 at least one SSC post will have more than 100,000 hits - Alexei Andreev
- In 2017 there will be no major intifada in Israel (ie > 250 Israeli deaths, but not in Cast Lead style war) - Alexei Andreev
- In 2017, Assad will remain President of Syria - Alexei Andreev
- In 2017, EMDrive is launched into space and testing is successfully begun - Alexei Andreev
- In 2017, Israel will not get in a large-scale war (ie >100 Israeli deaths) with any Arab state - Alexei Andreev
- In 2017, North Korea’s government will survive the year without large civil war/revolt - Alexei Andreev
- In 2017, Trump administration will not initiate extra prosecution of Hillary Clinton - Alexei Andreev
- In 2017, US does not publicly and explicitly disavow One China policy - Alexei Andreev
- In 2017, US does not withdraw from large trade org like WTO or NAFTA - Alexei Andreev
- In 2017, US lifts at least half of existing sanctions on Russia - Alexei Andreev
- In 2017, US will not get involved in any new major war with death toll of > 100 US soldiers - Alexei Andreev
- In 2017, Ukraine will neither break into all-out war or get neatly resolved - Alexei Andreev
- In 2017, a significant number of believers will not become convinced EMDrive doesn’t work - Alexei Andreev
- In 2017, a significant number of skeptics will not become convinced EMDrive works - Alexei Andreev
- In 2017, construction on Mexican border wall (beyond existing barriers) begins - Alexei Andreev
- In 2017, no terrorist attack in any First World country will kill > 100 people - Alexei Andreev
- In 2017, no terrorist attack in the USA will kill > 100 people - Alexei Andreev
- In 2017, there will be no major civil war in Middle Eastern country not already experiencing a major civil war at the beginning of 2017 - Alexei Andreev
- Keith Ellison will be chosen as new DNC chair in 2017 - Alexei Andreev
- Less Wrong renaissance attempt will seem less (rather than more) successful by end of 2017 - Alexei Andreev
- Libya will remain a mess in 2017 - Alexei Andreev
- Marine Le Pen will not be elected President of France in 2017 - Alexei Andreev
- No agreement will be reached on "two-speed EU" - Alexei Andreev
- No country currently in Euro or EU will announce new plan to leave in 2017 - Alexei Andreev
- No exchange of fire over "tiny stupid islands" in 2017 - Alexei Andreev
- No major earthquake (>10,000 deaths) in the world in 2017 - Alexei Andreev
- No major earthquake (>100 deaths) in US in 2017 - Alexei Andreev
- No major war in Asia (with >100 Chinese, Japanese, South Korean, and American deaths combined) over "tiny stupid islands" in 2017 - Alexei Andreev
- No serious impeachment proceedings are active against Trump in 2017 - Alexei Andreev
- Oil will end 2017 higher than $50 a barrel - Alexei Andreev
- Oil will end 2017 lower than $60 a barrel - Alexei Andreev
- SSC will remain active by end of 2017 - Alexei Andreev
- Shanghai index will not end 2017 down by more than 10% - Alexei Andreev
- Situation in Israel will look more worse than better by end of 2017 - Alexei Andreev
- SlateStarCodex will have more than 15,000 Twitter followers by end of 2017 - Alexei Andreev
- Syria’s civil war will not end in 2017 - Alexei Andreev
- The UK will trigger Article 50 in 2017 - Alexei Andreev
- There will be no Cast Lead style bombing/invasion of Gaza in 2017 - Alexei Andreev
- There will be no announcement of genetically engineered human baby or credible plan for such in 2017 - Alexei Andreev
- There will be no race riot killing > 5 people in 2017 - Alexei Andreev
- Theresa May will remain PM of Britain in 2017 - Alexei Andreev
- US unemployment to be higher at end of 2017 than beginning - Alexei Andreev

## Project proposal: Intro to the Universal Property

- Project outline: Intro to the Universal Property
Outline detailing all the work required for a proposed Arbital Project

- Eric Rogstad

## Rationality

- Epistemology
What is truth?

- Eliezer Yudkowsky- Central examples
The "central examples" for a subject are examples that are referred to over and over again in the co…

- Eliezer Yudkowsky - Conceivability
A hypothetical scenario is 'conceivable' or 'imaginable' when it is not *immediately* incoherent, al…

- Eliezer Yudkowsky - Fallacies
To call something a fallacy is to assert that you think people shouldn't think like that.

- Eliezer Yudkowsky- Bulverism
Bulverism is when you explain what goes so horribly wrong in people's minds when they believe X, before you've actually explained why X is wrong. Forbidden on Arbital.

- Eliezer Yudkowsky - Emphemeral premises
When somebody says X, don't just say, "Oh, not-X because Y" and then forget about Y a day later. Y is now an important load-bearing assumption in your worldview. Write Y down somewhere.

- Eliezer Yudkowsky - Gotcha button
A conversational point which, when pressed, causes the other person to shout "Gotcha!" and leap on what they think is a weakness allowing them to dismiss the conversation.

- Eliezer Yudkowsky - Harmless supernova fallacy
False dichotomies and continuum fallacies which can be used to argue that anything, including a supernova, must be harmless.

- Eliezer Yudkowsky - Invisible background fallacies
Universal laws also apply to objects and ideas that may fade into the invisible background. Reasoning as if these laws didn't apply to less obtrusive concepts is a type of fallacy.

- Eliezer Yudkowsky - Mind projection fallacy
Uncertainty is in the mind, not in the environment; a blank map does not correspond to a blank territory. In general, the territory may have a different ontology from the map.

- Eliezer Yudkowsky - Multiple stage fallacy
You can make an arbitrary proposition sound very improbable by observing how it seemingly requires X, Y, and Z. This didn't work for Nate Silver forecasting the Trump nomination.

- Eliezer Yudkowsky - Proving too much
If your argument could just as naturally be used to prove that Bigfoot exists and that Peano arithmetic is inconsistent, maybe it's an untrustworthy kind of argument.

- Eliezer Yudkowsky

- Bulverism
- Guarded definition
A guarded definition is one where at least one position suspects there will be pressure to stretch a…

- Eliezer Yudkowsky - Intension vs. extension
"Red is a light with a wavelength of 700 nm" vs. "Look at this red apple, red car, and red cup."

- Eliezer Yudkowsky - Perfect rolling sphere
If you don't understand something, start by assuming it's a perfect rolling sphere.

- Eliezer Yudkowsky - Psychologizing
It's sometimes important to consider how other people might be led into error. But psychoanalyzing them is also dangerous! Arbital discussion norms say to explicitly note this as "psychologizing".

- Eliezer Yudkowsky- Bulverism
Bulverism is when you explain what goes so horribly wrong in people's minds when they believe X, before you've actually explained why X is wrong. Forbidden on Arbital.

- Eliezer Yudkowsky

- Bulverism
- Strained argument
A phenomenological feeling associated with a step of reasoning going from X to Y where it feels like…

- Eliezer Yudkowsky - Strictly factual question
A "question of strict fact" is one which is true or false about the material universe (and maybe some math) without introducing any issues of values, perspectives, etcetera.

- Eliezer Yudkowsky - The empiricist-theorist false dichotomy
No discussion here yet: See https://www.facebook.com/groups/674486385982694/permalink/7846641016315…

- Eliezer Yudkowsky

- Central examples
- Humans doing Bayes
The human use of Bayesian reasoning in everyday life

- Eliezer Yudkowsky- Explicit Bayes as a counter for 'worrying'
Explicitly walking through Bayes's Rule can summarize your knowledge and thereby stop you from bouncing around pieces of it.

- Eliezer Yudkowsky

- Explicit Bayes as a counter for 'worrying'
- Ideological Turing test
Can you explain the opposing position well enough that people can't tell whether you or a real advocate of that position created the explanation?

- Eliezer Yudkowsky - Introduction to Effective Altruism
Effective altruism (EA) means using evidence and reason to take actions that help others as much as …

- Aaron Gertler - Probability theory
The logic of science; coherence relations on quantitative degrees of belief.

- Eliezer Yudkowsky- Bayesian reasoning
A probability-theory-based view of the world; a coherent way of changing probabilistic beliefs based on evidence.

- Eliezer Yudkowsky- Bayes' rule
Bayes' rule is the core theorem of probability theory saying how to revise our beliefs when we make a new observation.

- Eliezer Yudkowsky- Bayes' rule examples
Interesting problems solvable by Bayes' rule

- Eliezer Yudkowsky- Introductory Bayesian problems
Bayesian problems to try to solve yourself, before beginning to learn about Bayes' rule.

- Eliezer Yudkowsky- Blue oysters
A probability problem about blue oysters.

- Nate Soares - Diseasitis
20% of patients have Diseasitis. 90% of sick patients and 30% of healthy patients turn a tongue depressor black. You turn a tongue depressor black. What's the chance you have Diseasitis?

- Eliezer Yudkowsky - Sock-dresser search
There's a 4/5 chance your socks are in one of your dresser's 8 drawers. You check 6 drawers at random. What's the probability they'll be in the next drawer you check?

- Nate Soares - Sparking widgets
10% of widgets are bad and 90% are good. 4% of good widgets emit sparks, and 12% of bad widgets emit…

- Nate Soares

- Blue oysters
- Realistic (Math 1)
Real-life examples of Bayesian reasoning

- Eliezer Yudkowsky

- Introductory Bayesian problems
- Bayes' rule: Beginner's guide
Beginner's guide to learning about Bayes' rule.

- Alexei Andreev - Bayes' rule: Definition
Bayes' rule is the mathematics of probability theory governing how to update your beliefs in the lig…

- Nate Soares - Bayes' rule: Functional form
Bayes' rule for to continuous variables.

- Eliezer Yudkowsky - Bayes' rule: Guide
The Arbital guide to Bayes' rule

- Eliezer Yudkowsky- Bayes' Rule and its different forms
This is an arc that includes different ways to look at Bayes' Rule.

- Alexei Andreev - Bayes' Rule and its implications
This is an arc that includes implications of the Bayes's Rule.

- Alexei Andreev - Comprehensive guide to Bayes' Rule
This is an arc that includes all Bayes content.

- Alexei Andreev - Introduction to Bayes' Rule odds form
This is an arc that includes just enough content to teach about Bayes's Rule odds form.

- Alexei Andreev - Wants to get straight to Bayes
A simple requisite page to mark whether the user has selected wanting to get straight into Bayes on …

- Eliezer Yudkowsky

- Bayes' Rule and its different forms
- Bayes' rule: Log-odds form
A simple transformation of Bayes' rule reveals tools for measuring degree of belief, and strength of evidence.

- Eliezer Yudkowsky - Bayes' rule: Odds form
The simplest and most easily understandable form of Bayes' rule uses relative odds.

- Eliezer Yudkowsky- Introduction to Bayes' rule: Odds form
Bayes' rule is simple, if you think in terms of relative odds.

- Eliezer Yudkowsky

- Introduction to Bayes' rule: Odds form
- Bayes' rule: Probability form
The original formulation of Bayes' rule.

- Nate Soares- Odds form to probability form
The odds form of Bayes' rule works for any two hypotheses $H_i$ and $H_j,$ and looks like this: $$\…

- Nate Soares - Proof of Bayes' rule: Probability form
Let $\mathbf H$ be a [random\_variable variable] in $\mathbb P$ for the true hypothesis, and let $H_…

- Nate Soares

- Odds form to probability form
- Bayes' rule: Proportional form
The fastest way to say something both convincing and true about belief-updating.

- Eliezer Yudkowsky - Bayes' rule: Vector form
For when you want to apply Bayes' rule to lots of evidence and lots of variables, all in one go. (This is more or less how spam filters work.)

- Eliezer Yudkowsky - Belief revision as probability elimination
Update your beliefs by throwing away large chunks of probability mass.

- Eliezer Yudkowsky - Frequency diagram
Visualizing Bayes' rule by manipulating frequencies in large populations

- Nate Soares- Frequency diagrams: A first look at Bayes
The most straightforward visualization of Bayes' rule.

- Nate Soares

- Frequency diagrams: A first look at Bayes
- High-speed intro to Bayes's rule
A high-speed introduction to Bayes's Rule on one page, for the impatient and mathematically adept.

- Eliezer Yudkowsky - Path: Multiple angles on Bayes's Rule
A learning-path placeholder page for learning multiple angles on Bayes's Rule.

- Eliezer Yudkowsky - Probability notation for Bayes' rule
The probability notation used in Bayesian reasoning

- Eliezer Yudkowsky- Probability notation for Bayes' rule: Intro (Math 1)
How to read, and identify, the probabilities in Bayesian problems.

- Eliezer Yudkowsky

- Probability notation for Bayes' rule: Intro (Math 1)
- Proof of Bayes' rule
Proofs of Bayes' rule, with graphics

- Eliezer Yudkowsky- Proof of Bayes' rule: Probability form

- Proof of Bayes' rule: Probability form
- Shift towards the hypothesis of least surprise
When you see new evidence, ask: which hypothesis is *least surprised?*

- Nate Soares - Waterfall diagram
Visualizing Bayes' rule as the mixing of probability streams.

- Eliezer Yudkowsky- Waterfall diagrams and relative odds
A way to visualize Bayes' rule that yields an easier way to solve some problems

- Eliezer Yudkowsky

- Waterfall diagrams and relative odds

- Bayes' rule examples
- Bayesian update
Bayesian updating: the ideal way to change probabilistic beliefs based on evidence.

- Eliezer Yudkowsky- Bayesian view of scientific virtues
Why is it that science relies on bold, precise, and falsifiable predictions? Because of Bayes' rule, of course.

- Eliezer Yudkowsky - Extraordinary claims require extraordinary evidence
The people who adamantly claim they were abducted by aliens do provide some evidence for aliens. They just don't provide quantitatively enough evidence.

- Eliezer Yudkowsky- Extraordinary claims
What makes something an 'extraordinary claim' that requires extraordinary evidence?

- Eliezer Yudkowsky

- Extraordinary claims
- Ordinary claims require ordinary evidence
Extraordinary claims require extraordinary evidence, but ordinary claims *don't*.

- Nate Soares - Path: Insights from Bayesian updating
A learning-path placeholder page for insights derived from the Bayesian rule for updating beliefs.

- Eliezer Yudkowsky - Strength of Bayesian evidence
From a Bayesian standpoint, the strength of evidence can be identified with its likelihood ratio.

- Eliezer Yudkowsky

- Bayesian view of scientific virtues
- Empirical probabilities are not exactly 0 or 1
"Cromwell's Rule" says that probabilities of exactly 0 or 1 should never be applied to empirical propositions - there's always some probability, however tiny, of being mistaken.

- Eliezer Yudkowsky - Finishing your Bayesian path on Arbital
The page that comes at the end of reading the Arbital Guide to Bayes' rule

- Eliezer Yudkowsky - Humans doing Bayes
The human use of Bayesian reasoning in everyday life

- Eliezer Yudkowsky- Explicit Bayes as a counter for 'worrying'
Explicitly walking through Bayes's Rule can summarize your knowledge and thereby stop you from bouncing around pieces of it.

- Eliezer Yudkowsky

- Explicit Bayes as a counter for 'worrying'
- Ignorance prior
Key equations for quantitative Bayesian problems, describing exactly the right shape for what we believed before observation.

- Eliezer Yudkowsky- Inductive prior
Some states of pre-observation belief can learn quickly; others never learn anything. An "inductive prior" is of the former type.

- Eliezer Yudkowsky- Laplace's Rule of Succession
Suppose you flip a coin with an unknown bias 30 times, and see 4 heads and 26 tails. The Rule of Succession says the next flip has a 5/32 chance of showing heads.

- Eliezer Yudkowsky - Solomonoff induction
A simple way to superintelligently predict sequences of data, given unlimited computing power.

- Eliezer Yudkowsky- Solomonoff induction: Intro Dialogue (Math 2)
An introduction to Solomonoff induction for the unfamiliar reader who isn't bad at math

- Eliezer Yudkowsky

- Solomonoff induction: Intro Dialogue (Math 2)
- Universal prior
A "universal prior" is a probability distribution containing *all* the hypotheses, for some reasonable meaning of "all". E.g., "every possible computer program that computes probabilities".

- Eliezer Yudkowsky

- Laplace's Rule of Succession

- Inductive prior
- Interest in mathematical foundations in Bayesianism
"Want" this requisite if you prefer to see extra information about the mathematical foundations in Bayesianism.

- Eliezer Yudkowsky - Likelihood
"Likelihood", when speaking of Bayesian reasoning, denotes *the probability of an observation, sup…

- Nate Soares- Likelihood function
Let's say you have a piece of evidence $e$ and a set of hypotheses $\mathcal H.$ Each $H_i \in \math…

- Nate Soares - Likelihood notation
The likelihood of a piece of evidence $e$ according to a hypothesis $H,$ known as "the likelihood of…

- Nate Soares - Likelihood ratio
Given a piece of evidence $e$ and two hypothsese $H_i$ and $H_j,$ the likelihood ratio between them…

- Nate Soares - Relative likelihood
How relatively likely an observation is, given two or more hypotheses, determines the strength and direction of evidence.

- Eliezer Yudkowsky

- Likelihood function
- Posterior probability
What we believe, after seeing the evidence and doing a Bayesian update.

- Eliezer Yudkowsky - Prior
A state of prior knowledge, before seeing information on a new problem. Potentially complicated.

- Eliezer Yudkowsky - Prior probability
What we believed before seeing the evidence.

- Eliezer Yudkowsky - Strictly confused
A hypothesis is strictly confused by the raw data, if the hypothesis did much worse in predicting it than the hypothesis itself expected.

- Eliezer Yudkowsky - Subjective probability
Probability is in the mind, not in the environment. If you don't know whether a coin came up heads or tails, that's a fact about you, not a fact about the coin.

- Eliezer Yudkowsky- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Nate Soares

- Probability distribution: Motivated definition

- Bayes' rule
- Mutually exclusive and exhaustive
The condition needed for probabilities to sum to 1

- Eliezer Yudkowsky - Normalization (probability)
That thingy we do to make sure our probabilities sum to 1, when they should sum to 1.

- Eliezer Yudkowsky - Odds
Odds express a relative probability.

- Eliezer Yudkowsky- Odds: Introduction
What's the difference between probabilities and odds? Why is a 20% probability of success equivalent to 1 : 4 odds favoring success?

- Nate Soares - Odds: Refresher
A quick review of the notations and mathematical behaviors for odds (e.g. odds of 1 : 2 for drawing a red ball vs. green ball from a barrel).

- Nate Soares - Odds: Technical explanation
Formal definitions, alternate representations, and uses of odds and odds ratios (like a 1 : 2 chance of drawing a red ball vs. green ball from a barrel).

- Alexei Andreev

- Odds: Introduction
- Probability
The degree to which someone believes something, measured on a scale from 0 to 1, allowing us to do math to it.

- Eliezer Yudkowsky- Conditional probability
The notation for writing "The probability that someone has green eyes, if we know that they have red hair."

- Eliezer Yudkowsky- Conditional probability: Refresher
Is P(yellow | banana) the probability that a banana is yellow, or the probability that a yellow thing is a banana?

- Nate Soares

- Conditional probability: Refresher
- Interpretations of "probability"
What does it *mean* to say that a fair coin has a 50% probability of coming up heads?

- Nate Soares- Correspondence visualizations for different interpretations of "probability"
Let's say you have a model which says a particular coin is 70% likely to be heads. How should we as…

- Nate Soares - Probability interpretations: Examples
Consider evaluating, in June of 2016, the question: "What is the probability of Hillary Clinton wi…

- Nate Soares - Subjective probability
- Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Nate Soares

- Probability distribution: Motivated definition

- Correspondence visualizations for different interpretations of "probability"
- Joint probability
The notation for writing the chance that both X and Y are true.

- Eliezer Yudkowsky - Report likelihoods, not p-values
If scientists reported likelihood functions instead of p-values, this could help science avoid p-ha…

- Nate Soares- Likelihood functions, p-values, and the replication crisis
What's the whole Bayesian-vs.-frequentist debate about?

- Eliezer Yudkowsky - Report likelihoods not p-values: FAQ
This page answers frequently asked questions about the Report likelihoods, not p-values proposal for…

- Nate Soares

- Likelihood functions, p-values, and the replication crisis

- Conditional probability
- Probability distribution (countable sample space)
A function assigning a probability to each point in the sample space.

- Tsvi BT - Probability distribution: Motivated definition
People keep writing things like P(sick)=0.3. What does this mean, on a technical level?

- Nate Soares - Sample space
The set of possible things that could happen in a part of the world that you are uncertain about.

- Tsvi BT- Sample spaces are too large
Sample spaces are often large, so it is hard to do probabilistic computations using a raw distribution over the sample space.

- Tsvi BT - Uncountable sample spaces are way too large
We can't define probability distributions over uncountable sample spaces by just assigning numbers to each point in the sample space.

- Tsvi BT

- Sample spaces are too large
- Square visualization of probabilities on two events
$$ \newcommand{\true}{\text{True}} \newcommand{\false}{\text{False}} \newcommand{\bP}{\mathbb{P}} …

- Tsvi BT- Square visualization of probabilities on two events: (example) Diseasitis
But it *seems* like the patient with the black tongue depressor has diseasitis...

- Tsvi BT

- Square visualization of probabilities on two events: (example) Diseasitis
- Two independent events
What do [a pair of dice], [a pair of coins], and [a pair of people on opposite sides of the planet] all have in common?

- Tsvi BT- Two independent events: Square visualization
$$ \newcommand{\true}{\text{True}} \newcommand{\false}{\text{False}} \newcommand{\bP}{\mathbb{P}} …

- Tsvi BT

- Two independent events: Square visualization
- Well-calibrated probabilities
Even if you're fairly ignorant, you can still strive to ensure that when you say "70% probability", it's true 70% of the time.

- Eliezer Yudkowsky

- Bayesian reasoning

## The plan experiment

- AI arms race
This part of the plan describes how to prevent and handle AI arms race.

- Alexei Andreev - Organizations and people
Comprehensive list of organizations and people relevant to navigating development of AGI.

- Alexei Andreev - Policy desiderata in the development of machine superintelligence
Link to the paper by Nick Bostrom, Allan Dafoe, and Carrick Flynn.

- Alexei Andreev - Stages
List of stages in AGI development and their properties.

- Alexei Andreev - The plan
Root page for the plan on how to approach and navigate through AGI development.

- Alexei Andreev - The plan should be public
There is an open question of whether or not The plan experiment should be public. My current intuiti…

- Alexei Andreev - What should superintelligence be programmed to do?
This is one of the key open questions in The plan experiment. Coherent extrapolated volition (align…

- Alexei Andreev

## Vitamin D

- Calcitriol
Calcitriol, also called 1,25-dihydroxycholecalciferol or 1,25-dihydroxyvitamin D<sub>3</sub>, is the…

- Alexei Andreev- Calcitriol in the Treatment of Prostate Cancer
Review Link: http://ar.iiarjournals.org/content/26/4A/2647.full.pdf From the abstract: > "Calcitr…

- Alexei Andreev

- Calcitriol in the Treatment of Prostate Cancer
- Calcium Plus Vitamin D Supplementation and the Risk of Breast Cancer
Randomized trial Link: http://jnci.oxfordjournals.org/content/100/22/1581.long From conclusions: …

- Alexei Andreev - Calcium plus Vitamin D Supplementation and the Risk of Colorectal Cancer
Randomized trial Link: http://www.fp.ucalgary.ca/FMResidentSecure/Articles/Ca%20vit%20d%20and%20ca%…

- Alexei Andreev - Cholecalciferol
Cholecalciferol (toxiferol, vitamin D<sub>3</sub>) is one of the five forms of vitamin D. It is a se…

- Alexei Andreev- Effect of four monthly oral vitamin D3 (cholecalciferol) supplementation on fractures and mortality in men and women living in the community: randomised double blind controlled trial
Randomised double blind controlled trial. Link: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC150177/…

- Alexei Andreev

- Effect of four monthly oral vitamin D3 (cholecalciferol) supplementation on fractures and mortality in men and women living in the community: randomised double blind controlled trial
- Does solar exposure, as indicated by the non-melanoma skin cancers, protect from solid cancers: Vitamin D as a possible explanation
Correlational study Link: http://www.ejcancer.com/article/S0959-8049%2807%2900324-3/abstract From …

- Alexei Andreev - Molecular basis of the potential of vitamin D to prevent cancer
Theoretical meta-analysis Link: http://www.researchgate.net/publication/5812584_Molecular_basis_of_…

- Alexei Andreev - Serum 25-Hydroxyvitamin D Levels and the Prevalence of Peripheral Arterial Disease
Clinical and population study Link: http://atvb.ahajournals.org/content/28/6/1179.full From conclu…

- Alexei Andreev - Serum 25-Hydroxyvitamin D and Risks of Colon and Rectal Cancer in Finnish Men
Case-control study. Link: http://aje.oxfordjournals.org/content/173/5/499.full From discussion: >…

- Alexei Andreev - Serum Vitamin D Concentration and Prostate Cancer Risk: A Nested Case – Control Study
Prospective analysis. Link: http://jnci.oxfordjournals.org/content/100/11/796.full.pdf From conclu…

- Alexei Andreev - The effect of vitamin D supplementation on skeletal, vascular, or cancer outcomes: a trial sequential meta-analysis
Trial sequential meta-analysis. Link: http://www.thelancet.com/journals/landia/article/PIIS2213-858…

- Alexei Andreev - Vitamin D Deficiency and Risk for Cardiovascular Disease
Theoretical meta-analysis. Link: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2726624/ From conclus…

- Alexei Andreev - Vitamin D and Cardiometabolic Outcomes: A Systematic Review
Meta-analysis. Link: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3211092/ From conclusions: > "An…

- Alexei Andreev - Vitamin D and calcium supplementation reduces cancer risk: results of a randomized trial
Randomized trial Link: http://www.ncbi.nlm.nih.gov/pubmed/17556697%20 From conclusions: > "Improv…

- Alexei Andreev - Vitamin D helps prevent bone fracture
(Yes) One randomized trial shows some promise, but at extremely high doses of 100,000 IUs. (No) A l…

- Alexei Andreev - Vitamin D helps prevent breast cancer
No -- Calcium and vitamin D supplementation did not reduce invasive breast cancer incidence in post…

- Alexei Andreev - Vitamin D helps prevent cancer
There have been a lot of studies performed that show that vitamin D helps prevent cancer, but overal…

- Alexei Andreev - Vitamin D helps prevent cardiovascular disease
This question is a bit vague. There are two ways to make it more specific. 1. Does vitamin D deffic…

- Alexei Andreev - Vitamin D helps prevent colorectal cancer
No -- Daily supplementation of calcium with vitamin D for seven years had no effect on the incidenc…

- Alexei Andreev - Vitamin D helps prevent prostate cancer
Yes --- Calcitriol showed significant antineoplastic activity in pre-clinical models of prostate ca…

- Alexei Andreev - Vitamin D is good for you
We'll consider two categories of vitamin D supplementation: below and above the recommended levels. …

- Alexei Andreev - Vitamin D is related to blood pressure and other cardiovascular risk factors in middle-aged men
Correlational study Link: http://www.ncbi.nlm.nih.gov/pubmed/8541004

- Alexei Andreev - Vitamin, Mineral, and Multivitamin Supplements for the Primary Prevention of Cardiovascular Disease and Cancer: A Systematic Evidence Review for the US Preventive Services Task Force
Meta-study Link: http://www.ncbi.nlm.nih.gov/pubmed/24308073 From results: > "Vitamin D and/or ca…

- Alexei Andreev

## Why waiting to donate harms charities

- Displaying the list of fundraiser donors sorted by the donation date would help with the "wait and see" problem. - Alexei Andreev
- Fundraisers should have a threshold amount which, if not hit, results in a refund.
When starting a fundraiser, a nonprofit should declare a threshold amount. If the nonprofit doesn't …

- Alexei Andreev - I often wait to see how much other people will donate to a fundraiser before donating myself. - Alexei Andreev
- It's good for GiveWell and Good Ventures to crowd out donors by their donations. - Alexei Andreev
- When I donate to a charity, I am concerned whether or not the charity will raise enough money to make my donation worthwhile. - Alexei Andreev