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text: '[summary: The arithmetical hierarchy classifies logical statements by the number of nested clauses saying "for every object" and "there exists an object". Statements with one "for every object" clause belong in $\\Pi_1$, and statements with one "there exists an object" clause belong in $\\Sigma_1$. Saying "There exists an object x such that (some $\\Pi_n$ statement treating x as a constant)" creates a $\\Sigma_{n+1}$ statement. Similarly, adding a "For every x" clause outside a $\\Sigma_n$ statement creates a $\\Pi_{n+1}$ statement. Statements that can be formulated in both $\\Pi_n$ and $\\Sigma_n$ are said to lie in $\\Delta_n$. Some interesting consequences are that $\\Pi_1$ statements are falsifiable by observation, $\\Sigma_1$ statements are verifiable by observation, and statements strictly in higher classes can only be probabilistically verified by observation.]\n\n[summary(Technical): The arithmetical hierarchy classifies statements by the number of nested, unbounded quantifiers they contain. The classes $\\Delta_0$, $\\Pi_0$, and $\\Sigma_0$ are equivalent and include statements containing only bounded quantifiers, e.g. $\\forall x < 10: \\exists y < x: x + y < 10$. If, treating $x, y, z...$ as constants, a statement $\\phi(x, y, z...)$ would be in $\\Sigma_n,$ then adjoining the unbounded universal quantifiers $\\forall x: \\forall y: \\forall z: ... \\phi(x, y, z...)$ creates a $\\Pi_{n+1}$ statement. Similarly, adjoining existential quantifiers to a $\\Pi_n$ statement creates a $\\Sigma_{n+1}$ statement. Statements that can be equivalently formulated to be in both $\\Pi_n$ and $\\Sigma_n$ are said to lie in $\\Delta_n$. Interesting consequences include, e.g., $\\Pi_1$ statements are falsifiable by simple observation, $\\Sigma_1$ statements are verifiable by observation, and statements strictly in higher classes can only be probabilistically verified by observation.]\n\nThe arithmetical hierarchy classifies statements according to the number of unbounded $\\forall x$ and $\\exists y$ quantifiers, treating adjacent quantifiers of the same type as a single quantifier.\n\nThe formula $\\phi(x, y) \\leftrightarrow [(x + y) = (y + x)],$ treating $x$ and $y$ as constants, contains no quantifiers and would occupy the lowest level of the hierarchy, $\\Delta_0 = \\Pi_0 = \\Sigma_0.$ (Assuming that the operators $+$ and $=$ are themselves considered to be in $\\Delta_0$, or from another perspective, that for any particular $c$ and $d$ we can verify whether $c + d = d + c$ in bounded time.)\n\nAdjoining any number of $\\forall x_1: \\forall x_2: ...$ quantifiers to a statement that would be in $\\Sigma_n$ if the $x_i$ were considered as constants, creates a statement in $\\Pi_{n+1}.$ Thus, the statement $\\forall x: (x + 3) = (3 + x)$ is in $\\Pi_1.$\n\nSimilarly, adjoining $\\exists x_1: \\exists x_2: ...$ to a statement in $\\Pi_n$ creates a statement in $\\Sigma_{n+1}.$ Thus, the statement $\\exists y: \\forall x: (x + y) = (y + x)$ is in $\\Sigma_2$, while the statement $\\exists y: \\exists x: (x + y) = (y + x)$ is in $\\Sigma_1.$\n\nStatements in both $\\Pi_n$ and $\\Sigma_n$ (e.g. because they have provably equivalent formulations belonging to both classes) are said to lie in $\\Delta_n.$\n\nQuantifiers that can be bounded by $\\Delta_0$ functions of variables already introduced are ignored by this classification schema: the sentence $\\forall x: \\exists y < x: (x + y) = (y + x)$ is said to lie in $\\Pi_1$, not $\\Pi_2$. We can justify this by observing that for any particular $c,$ the statement $\\forall x < c: \\phi(x)$ can be expanded into the non-quantified statement $\\phi(0) \\wedge \\phi(1) ... \\wedge \\phi(c)$ and similarly $\\exists x < c: \\phi(x)$ expands to $\\phi(0) \\vee \\phi(1) \\vee ...$\n\nThis in turn justifies collapsing adjacent quantifiers of the same type inside the classification schema. Since, e.g., we can uniquely encode every pair (x, y) in a single number $z = 2^x \\cdot 3^y$, to say "there exists a pair (x, y)" or "for every pair (x, y)" it suffices to quantify over z encoding (x, y) with x and y less than z.\n\nWe say that $\\Delta_{n+1}$ includes the entire sets $\\Pi_n$ and $\\Sigma_n$, since from a $\\Pi_{n}$ statement we can produce a $\\Pi_{n+1}$ statement just by adding an inner $\\exists$ quantifier and then ignoring it, and we can obtain a $\\Sigma_{n+1}$ statement from a $\\Pi_{n}$ statement by adding an outer $\\forall$ quantifier and ignoring it, etcetera.\n\nThis means that the arithmetic hierarchy talks about *power sufficient to resolve statements*. To say $\\phi \\in \\Pi_n$ asserts that if you can resolve all $\\Pi_n$ formulas then you can resolve $\\phi$, which might potentially also be doable with less power than $\\Pi_n$, but can definitely not require more power than $\\Pi_n.$\n\n# Consequences for epistemic properties\n\nAll and only statements in $\\Sigma_1$ are *verifiable by observation*. If $\\phi \\in \\Delta_0$ then the sentence $\\exists x: \\phi(x)$ can be positively known by searching for and finding a single example. Conversely, if a statement involves an unbounded universal quantifier, we can never be sure of it through simple observation because we can't observe the truth for every possible number.\n\nAll and only statements in $\\Pi_1$ are *falsifiable by observation*. If $\\phi$ can be tested in bounded time, then we can falsify the whole statement $\\forall x: \\phi(x)$ by presenting some single x of which $\\phi$ is false. Conversely, if a statement involves an unbounded existential quantifier, we can never falsify it directly through a bounded number of observations because there could always be some higher, as-yet untested number that makes the sentence true.\n\nThis doesn't mean we can't get [1ly probabilistic confirmation and disconfirmation] of sentences outside $\\Sigma_1$ and $\\Pi_1.$ E.g. for a $\\Pi_2$ statement, "For every x there is a y", each time we find an example of a y for another x, we might become a little more confident, and if for some x we fail to find a y after long searching, we might become a little less confident in the entire statement.',
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