{ localUrl: '../page/axiom_of_choice.html', arbitalUrl: 'https://arbital.com/p/axiom_of_choice', rawJsonUrl: '../raw/69b.json', likeableId: '3564', likeableType: 'page', myLikeValue: '0', likeCount: '3', dislikeCount: '0', likeScore: '3', individualLikes: [ 'EricBruylant', 'NickBrown', 'JaimeSevillaMolina' ], pageId: 'axiom_of_choice', edit: '26', editSummary: 'pointed linearly ordered set link to current page at totally ordered set', prevEdit: '25', currentEdit: '26', wasPublished: 'true', type: 'wiki', title: 'Axiom of Choice', clickbait: 'The most controversial axiom of the 20th century.', textLength: '42774', alias: 'axiom_of_choice', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'EricBruylant', editCreatedAt: '2016-12-02 21:22:51', pageCreatorId: 'MarkChimes', pageCreatedAt: '2016-09-30 12:45:17', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '265', text: '[summary: The axiom of choice states that given an infinite collection of non-empty sets, there is a function that picks out one element from each set. ]\n\n"The Axiom of Choice is necessary to select a set from an infinite number of pairs of socks, but not an infinite number of pairs of shoes." — *Bertrand Russell, Introduction to mathematical philosophy*\n\n"Tarski told me the following story. He tried to publish his theorem \\[the equivalence between the Axiom of Choice and the statement 'every infinite set A has the same cardinality as AxA'\\] in the Comptes Rendus Acad. Sci. Paris but Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest. And Tarski said that after this misadventure he never again tried to publish in the Comptes Rendus."\n- *Jan Mycielski, A System of Axioms of Set Theory for the Rationalists*\n\n[toc:]\n\n#Obligatory Introduction#\nThe Axiom of Choice, the most controversial axiom of the 20th Century. \n\nThe axiom states that a certain kind of function, called a `choice' function, always exists. It is called a choice function, because, given a collection of non-empty sets, the function 'chooses' a single element from each of the sets. It is a powerful and useful axiom, asserting the existence of useful mathematical structures (such as bases for [-3w0 vector spaces] of arbitrary [-dimension_mathematics dimension], and [-ultraproduct ultraproducts]). It is a generally accepted axiom, and is in wide use by mathematicians. In fact, according to Elliott Mendelson in Introduction to Mathematical Logic (1964) "The status of the Axiom of Choice has become less controversial in recent years. To most mathematicians it seems quite plausible and it has so many important applications in practically all branches of mathematics that not to accept it would seem to be a wilful hobbling of the practicing mathematician. "\n\nNeverless, being an [-axiom_mathematics axiom], it cannot be proven and must instead be assumed. \nIn particular, it is an axiom of [-set_theory set theory] and it is not provable from the other axioms (the Zermelo-Fraenkel axioms of Set Theory). In fact many mathematicians, in particular [-constructive_mathematics constructive] mathematicians, reject the axiom, stating that it does not capture a 'real' or 'physical' property, but is instead just a mathematical oddity, an artefact of the mathematics used to approximate reality, rather than reality itself. In the words of the LessWrong community: the constructive mathematicians would claim it is a statement about [-https://wiki.lesswrong.com/wiki/Map_and_Territory_(sequence) the map, and not the territory]. \n\nHistorically, the axiom has experienced much controversy. Before it was shown to be independent of the other axioms, it was believed either to follow from them (i.e., be 'True') or lead to a contradiction (i.e., be 'False'). Its independence from the other axioms was, in fact, a very surprising result at the time. \n\n#Getting the Heavy Maths out the Way: Definitions#\nIntuitively, the [-axiom_mathematics axiom] of choice states that, given a collection of *[-5zc non-empty]* [-3jz sets], there is a [-3jy function] which selects a single element from each of the sets. \n\nMore formally, given a set $X$ whose [-5xy elements] are only non-empty sets, there is a function \n$$\nf: X \\rightarrow \\bigcup_{Y \\in X} Y \n$$\nfrom $X$ to the [-5s8 union] of all the elements of $X$ such that, for each $Y \\in X$, the [-3lh image] of $Y$ under $f$ is an element of $Y$, i.e., $f(Y) \\in Y$. \n\nIn [-logical_notation logical notation],\n$$\n\\forall_X \n\\left( \n\\left[\\forall_{Y \\in X} Y \\not= \\emptyset \\right] \n\\Rightarrow \n\\left[\\exists \n\\left( f: X \\rightarrow \\bigcup_{Y \\in X} Y \\right)\n\\left(\\forall_{Y \\in X} \n\\exists_{y \\in Y} f(Y) = y \\right) \\right]\n\\right)\n$$\n\n#Axiom Unnecessary for Finite Collections of Sets#\nFor a [-5zy finite set] $X$ containing only [-5zy finite] non-empty sets, the axiom is actually provable (from the [-zermelo_fraenkel_axioms Zermelo-Fraenkel axioms] of set theory ZF), and hence does not need to be given as an [-axiom_mathematics axiom]. In fact, even for a finite collection of possibly infinite non-empty sets, the axiom of choice is provable (from ZF), using the [-axiom_of_induction axiom of induction]. In this case, the function can be explicitly described. For example, if the set $X$ contains only three, potentially infinite, non-empty sets $Y_1, Y_2, Y_3$, then the fact that they are non-empty means they each contain at least one element, say $y_1 \\in Y_1, y_2 \\in Y_2, y_3 \\in Y_3$. Then define $f$ by $f(Y_1) = y_1$, $f(Y_2) = y_2$ and $f(Y_3) = y_3$. This construction is permitted by the axioms ZF.\n\nThe problem comes in if $X$ contains an infinite number of non-empty sets. Let's assume $X$ contains a [-2w0 countable] number of sets $Y_1, Y_2, Y_3, \\ldots$. Then, again intuitively speaking, we can explicitly describe how $f$ might act on finitely many of the $Y$s (say the first $n$ for any natural number $n$), but we cannot describe it on all of them at once. \n\nTo understand this properly, one must understand what it means to be able to 'describe' or 'construct' a function $f$. This is described in more detail in the sections which follow. But first, a bit of background on why the axiom of choice is interesting to mathematicians.\n\n#Controversy: Mathematicians Divided! Counter-Intuitive Results, and The History of the Axiom of Choice#\nMathematicians have been using an intuitive concept of a set for probably as long as mathematics has been practiced. \nAt first, mathematicians assumed that the axiom of choice was simply true (as indeed it is for finite collections of sets). \n\n[-https://en.wikipedia.org/wiki/Georg_Cantor Georg Cantor] introduced the concept of [-transfinite_number transfinite numbers] \nand different [-4w5 cardinalities of infinity] in a 1874 \n[https://en.wikipedia.org/wiki/Georg_Cantor%27s_first_set_theory_article paper] (which contains his infamous\n[-https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument Diagonalization Argument]) \n and along with this sparked the introduction of [-set_theory set theory].\n In 1883, Cantor introduced a principle called the 'Well-Ordering Princple'\n(discussed further in a section below) which he called a 'law of thought' (i.e., intuitively true). \nHe attempted to prove this principle from his other principles, but found that he was unable to do so.\n\n[-https://en.wikipedia.org/wiki/Ernst_Zermelo Ernst Zermelo] attempted to \ndevelop an [-axiom_system axiomatic] treatment of set theory. He \n managed to prove the Well-Ordering Principle in 1904 by introducing a new principle: The Principle of Choice.\nThis sparked much discussion amongst mathematicians. In 1908 published a paper containing responses to this debate,\nas well as a new formulation of the Axiom of Choice. In this year, he also published his first version of \nthe set theoretic axioms, known as the [-https://en.wikipedia.org/wiki/Zermelo_set_theory Zermelo Axioms of Set Theory].\nMathematicians, [-https://en.wikipedia.org/wiki/Abraham_Fraenkel Abraham Fraenkel] and \n[-https://en.wikipedia.org/wiki/Thoralf_Skolem Thoralf Skolem] improved this system (independently of each other)\ninto its modern version, the [-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory\n Zermelo Fraenkel Axioms of Set Theory].\n\nIn 1914, [https://en.wikipedia.org/wiki/Felix_Hausdorff Felix Hausdorff] proved \n[https://en.wikipedia.org/wiki/Hausdorff_paradox Hausdorff's paradox]. The ideas\nbehind this proof were used in 1924 by [-https://en.wikipedia.org/wiki/Stefan_Banach\nStefan Banach] and [-https://en.wikipedia.org/wiki/Alfred_Tarski Alfred Tarski]\nto prove the more famous Banach-Tarski paradox (discussed in more detail below).\nThis latter theorem is often quoted as evidence of the falsehood of the axiom \nof choice.\n\nBetween 1935 and 1938, [-https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] proved that\nthe Axiom of Choice is consistent with the rest of the ZF axioms.\n\nFinally, in 1963, [-https://en.wikipedia.org/wiki/Paul_Cohen Paul Cohen] developed a revolutionary\nmathematical technique called [-forcing_mathematics forcing], with which he proved that the \naxiom of choice could not be proven from the ZF axioms (in particular, that the negation of AC\nis consistent with ZF). For this, and his proof of the consistency of the negation of the \n[-continuum_hypothesis Generalized Continuum Hypothesis] from ZF, he was awarded a fields medal\nin 1966.\n\nThis axiom came to be accepted in the general mathematical community, but was rejected by the\n[-constructive_mathematics constructive] mathematicians as being fundamentally non-constructive. \nHowever, it should be noted that in many forms of constructive mathematics, \nthere are *provable* versions of the axiom of choice.\nThe difference is that in general in constructive mathematics, exhibiting a set of non-empty sets\n(technically, in constructive set-theory, these should be 'inhabited' sets) also amounts to \nexhibiting a proof that they are all non-empty, which amounts to exhibiting an element for all\nof them, which amounts to exhibiting a function choosing an element in each. So in constructive \nmathematics, to even state that you have a set of inhabited sets requires stating that you have a choice\nfunction to these sets proving they are all inhabited.\n\nSome explanation of the history of the axiom of choice (as well as some of its issues)\ncan be found in the \npaper "100 years of Zermelo's axiom of choice: what was the problem with it?"\nby the constructive mathematician \n[-https://en.wikipedia.org/wiki/Per_Martin-L%C3%B6f Per Martin-Löf]\nat [-http://comjnl.oxfordjournals.org/content/49/3/345.full this webpage]. \n\n(Martin-Löf studied under [-https://en.wikipedia.org/wiki/Andrey_Kolmogorov Andrey Kolmogorov] of\n [-5v Kolmogorov complexity] and has made contributions to [-3qq information theory], \n[-statistics mathematical_statistics], and [-mathematical_logic mathematical_logic], including developing a form of \nintuitionistic [-3sz]).\n\nA nice timeline is also summarised on [-http://plato.stanford.edu/entries/axiom-choice/index.html#note-6\nThe Stanford Encyclopaedia of Logic].\n\n#So, What is this Choice Thing Good for Anyways?#\nThe Axiom of Choice is in common use by mathematicians in practice. Amongst its many applications are the following:\n\n###Non-Empty Products###\nThis is the statment that taking the mathematical [-product_of_sets product] of non-empty sets will always yield a non-empty set.\nConsider infinitely many sets $X_1, X_2, X_3, \\ldots$ indexed by the natural numbers.\n Then an element of the product $\\prod_{i \\in \\mathbb{N}} X_i$ is a family (essentially an infinite tuple) \nof the form $(x_1, x_2, x_3, \\ldots )$ where $x_1 \\in X_1$, $x_2 \\in X_2$ and so on. However, to select such a tuple in\nthe product amounts to selecting a single element from each of the sets. Hence, without the axiom of choice, it is\nnot provable that there are any elements in this product. \n\nNote, that without the axiom, it is not necessarily the case that the product is empty. It just isn't provable that\nthere are any elements. However, it is consistent that there exists some product that is empty. Also, bear in mind\nthat even without the axiom of choice, there are products of this form which are non-empty. It just can't be shown\nthat they are non-empty in general.\n\nHowever, it does seem somewhat counterintuitive that such a product be non-empty. In general, given two non-empty sets, \n$X_1$ and $X_2$, the product is at least as large as either of the sets. Adding another non-empty set $X_3$ usually makes\nthe product larger still. It may seem strange, then, that taking the limit of this procedure could result in an empty\nstructure. \n\n###Existence of a basis for any (esp. infinite dimensional) vector space###\nThis example is discussed in more detail in the next section.\n\nA [-3w0] can be intuitively thought of as a collection of [-vector vectors] which themselves can be thought of as arrows (or the information of a 'direction' and a 'magnitude'). \n\nA vector space has a number of 'directions' in which the vectors can point, referred to as its [-vector_space_dimension]. For a finite-dimensional vector space, it is possible to find a [-vector_space_basis basis] consisting of vectors. The property of a basis is that: \n\n - any vector in the space can be built up as a combination of the\n vectors in the basis \n - none of the vectors in the basis can be built up as a combination of the rest of the vectors in the basis.\n\nFor finite-dimensional vector spaces, such a basis is finite and can be found. However, for infinite-dimensional vector spaces, a way of finding a basis does not always exist. In fact, if the axiom of choice is false, then there is an infinite-dimensional vector space for which it is impossible to find a basis.\n\n###Brouwer's Fixed-Point Theorem: the existence of a fixed point for a function from a "nice" shape to itself###\nAny [-continuous_function continuous functions] $f: C \\rightarrow C$ from a [-closed_disk] $C$ onto itself has a [-fixed_point] $x_0$. (In full generality, $C$ may be any [-5xr] [-compact_mathematics compact] [-3jz]).\n\nIn other words, there is at least one point $x_0 \\in C$ such that $f(x_0) = x_0$.\n\nThis works also for rectangles, and even in multiple dimensions. Hence, if true for real world objects, this theorem has the following somewhat surprising consequences:\n\n - Take two identical sheets of graph with marked coordinates. Crumple up sheet B and place the crumpled ball on top of sheet A. Then, there is some coordinate $(x , y)$ (where $x$ and $y$ are [-4bc real numbers], not necessarily [-48l integers]), such that this coordinate on sheet B is directly above that coordinate on sheet A. \n - Take a cup of tea. Stir it and let it settle. There is some point of the tea which ends up in the same place it started.\n - Take a map of France. Place it on the ground in France. Take a pin. There is a point on the map through which you could stick the pin and the pin will also stick into the ground at the point represented on the map.\n\n###Existence of ultrafilters and hence ultraproducts###\nThe following example is somewhat technical. An attempt is made to describe it very roughly.\n\nGiven an indexing set $I$, and a collection of mathematical structures \n$(A_i)_{i \\in I}$\n(of a certain type) indexed by $I$. (For example, let $I$ be the [-45h natural numbers] $\\mathbb{N}$ and let each of the mathematical structures $A_n$ be numbered). \n\nAn [-ultrafilter] $\\mathcal{U}$ on $I$ is a collection of subsets of $I$ of a special type. Intuitively it should be thought of as a collection of `big subsets' of $I$. It is possible to form the set of all [-cofinite] subsets\nof $I$ without the axiom of choice, and $\\mathcal{U}$ should contain at least these. However, for mathematical reasons, $\\mathcal{U}$ should also contain 'as many sets as possible'. However, in order to do so, there are some 'arbitrary choices' that have to be made. This is where the axiom of choice comes in.\n\nOne of the applications of ultrafilters is [-ultraproduct ultraproducts]. For each subset $X \\subseteq I$ such that $X \\in \\mathcal{U}$ there is a subcollection\n$(A_i)_{i \\in X}$ of $(A_i)_{i \\in I}$. Call such a subcollection a "large collection". The ultraproduct $A$ is a structure that captures the properties of large collections of the $A_i$s, in the sense that a statement of [-first_order_logic] is true of the ultraproduct $A$ if and only if it is true of some large collection of the $A_i$s.\n\nNow, any statement that is either true for cofinitely many $A_i$s or false for cofinitely many $A_i$s will be true or false respectively for $A$. But what about the other statements? This is where the arbitrary choices come in. Each statement needs to be either true or false of $A$, and we use the axiom of choice to form an ultrafilter that does that for us.\n\nOne basic example of an application of ultrafilters is forming the [-nonstandard_real_numbers].\n\nFurther examples of applications of the axiom of choice may be found on\nthe Wikipedia page [-https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents here] \nand [-https://en.wikipedia.org/wiki/Axiom_of_choice#Results_requiring_AC_.28or_weaker_forms.29_but_weaker_than_it here]. \n\n#Physicists Hate Them! Find out How Banach and Tarski Make Infinity Dollars with this One Simple Trick! #\n\nOne of the most counter-intuitive consequences of the Axiom of Choice is the Banach-Tarski Paradox. It is a theorem provable using the Zermelo-Fraenkel axioms along with the axiom of choice.\n\nThis theorem was proven in a 1924 paper by [-https://en.wikipedia.org/wiki/Stefan_Banach Stefan Banach] and\n[-https://en.wikipedia.org/wiki/Alfred_Tarski Alfred Tarski].\n\nIntuitively, what the theorem says is that it is possible to take a ball, cut it into five pieces, rotate and shift these pieces and end up with two balls. \n\nNow, there are some complications, including the fact the pieces themselves are infinitely complex, and the have to pass through each other when they are being shifted. There is no way a practical implementation of this theorem could be developed. Nevertheless, that the volume of a ball could be changed just by cutting, rotating and shifting seems highly counter-intuitive. \n\nA suprisingly good video explanation in laymans terms by Vsauce can be found [-https://www.youtube.com/watch?v=s86-Z-CbaHA Here].\n\nThe video [-https://www.youtube.com/watch?v=ZUHWNfqzPJ8 \nInfinity shapeshifter vs. Banach-Tarski paradox] by Mathologor advertises itself as a\n prequel to the above video, which puts you 'in the mindset' of a mathematician, so-to-speak, and makes the\nresult a bit less surprising. \n\nThis theorem is the main counterexample used as evidence of the falsehood of the Axiom of Choice. If not taken as evidence of its falsehood, thsi is at least used as evidence of the counter-intuitiveness of AC. \n\nQ: What's an Anagram of Banach-Tarski? \nA: Banach-Tarski Banach Tarski\n\n#How Something Can Exist Without Actually Existing: The Zermelo Fraenkel Axioms and the Existence of a Choice Function#\nThe [-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo-Fraenkel axioms of Set Theory]\n(ZF)\nintroduce a fundamental concept, called a [-3jz set], and a [-3nt relation] between \nsets, called\nthe [-5xy element of] relation (usally written $\\in$) where $x \\in X$ should be interpreted as\n'$x$ is contained inside $X$ in the way that an item is contained inside a box'. There are then a number\nof [-axiom_mathematics axioms] imposed on these fundamental objects and this relation.\n\nWhat one must remember is that theorems derived from these axioms are merely statements of the form that \n'if one has a system which satisfies these laws (even if, for example $\\in$ is interpreted as something entirely \ndifferent from being contained inside something), then it must also satisfy the statements of the theorems'.\nHowever, its general use is to imagine sets as being something like boxes which contain mathematical objects. \nMoreover, almost any statement of mathematics can be stated in terms of sets, where the mathematical objects\nin question become sets of a certain kind. In this way, since the mathematical objects in question are set up\nto satisfy the axioms, then anything which can be derived from these axioms will also hold for the mathematical\nobjects.\n\nIn particular, a [-function_mathematics function] can be interpreted as a specific kind of set. In particular, \nit is a set of [-ordered_pair ordered pairs] (more generally, [-tuple ordered n-tuples], each of which can \nitself be interpreted as a specific kind of set) satisfying a specific property. \n\nThere are different ways of stating the same axioms (by separating or combining axioms, giving different\nformulations of the same axioms, or giving different axioms that are equivalent given the other axioms) hence\nwhat follows is only a specific formulation, namely, the \n[-https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory Zermelo-Fraenkel one from Wikipedia].\n\nThe axioms begin by stating that two sets are the same if they have the same elements. \nThen the axiom regularity states sets are well-behaved in a certain way that's not so\nimportant to us right now. \n\nNow comes the part that is important for our purposes: \nThe axiom schema of specification (actually a schema specifying \ninfinitely many axioms, but we can pretend it is\njust one axiom for now). This is an axiom asserting the *existence* of certain sets. \nIn a sense, it allows one to 'create' a new set out of an existing one. Namely,\ngiven a set $X$ and a statement $\\phi$ of [-first_order_logic first order logic]\n(a statement about sets of a specific, very formal form, \nand which uses only the $\\in$ symbol and the reserved symbols of logic), it is \npossible to create a set $\\{x \\in X : \\phi(x) \\}$ of all of the elements of\n$X$ for which the formula $\\phi$ is true. \n\nFor example, if we know (or assume) the set of all numbers $\\mathbb{N}$ exists, and \nwe have some way of formalising the statement '$x$ is an even number' as a first-order\nstatement $\\phi(x)$, then the set of all even numbers exists.\n\nAdditionally the axioms of pairing and union, axiom schema of replacement, \nand axiom of power set are all of the form "Given that some sets $A, B, C, \\ldots$ \nexist, then some other set $X$ also exists. The axiom of infinity simply states\nthat an infinite set with certain properties exists. \n\nNotice that all of the above are axioms. It is not expected that any of them be proven.\nThey are simply assumptions that you make about whichever system you\nwant to reason. Any theorems that you can prove from these axioms will then be true\nabout your system. However, mathematicians generally agree that these axioms capture\nour intuitive notion about how "sets" of objects (or even concepts)\n should behave, and about which sets we are allowed to reason (which sets 'exist').\nMost of these (except maybe the axiom of infinity, and even that one possibly) \nseem to apply to our world and seem to work fine.\n\nNow, the last axiom, the axiom of choice (or the well-ordering principle) asserts\nthat a certain kind of function exists. It\n cannot\nbe proven from the above. In other words, given a system that satisfies all of\nthe above, it cannot be assumed that the system also satisfies this axiom\n(nor in fact that it does *not* satisfy this axiom). That's all\nthere is to it, really. \n\nYet, mathematicians do disagree about this axiom, and whether it applies to our \nworld as well. Some mathematicians take a [-https://en.wikipedia.org/wiki/Platonism Platonic]\nview of mathematics, in which mathematical objects such as sets actually exist in some \nabstract realm, and for which the axiom of choice is either true or false, but we do\nnot know which. Others take a highly constructive view (in many cases motivated\nby realism and the ability of the mathematics to model the world) in which \neither the axiom of choice is false, or infinite sets do not exist in which \ncase the axiom of choice is provable and hence superfluous. Others take the view\nthat the axiom is not true or false, but merely useless, and that anything provable\nfrom it is meaningless. Many seem not to care: the axiom is convenient for the mathematics\nthey wish to do (whose application they are not much concerned about in any case) and\nhence they assume it without qualm. \n\n#How Something can be Neither True nor False:\nHow Could we Possibly Know that AC is Independent of ZF?#\nIt has been stated above multiple times that the Axiom of Choice is independent from the other\nZermelo-Fraenkel axioms. In other words, it can neither be proven nor disproven from those axioms.\nBut how can we possibly know this fact? \n\nThe answer lies in [-model_theory models]. However, these are not physical or even computational models,\nbut models of a more abstract kind. \n\nFor example, a model of [-3g8 group theory] is a specific [-3gd group],\nwhich itself can be characterized as a specific set. Now, notice that the axioms of group theory\nsay nothing about whether a given group is abelian (commutative) or not. It does not follow\nfrom the axioms that for groups it is always true that $xy = yx$, nor does it follow that for\ngroups there are always some $x$ and $y$ for which $xy \\not= yx$. In other words, the \n"abelian axiom" is independent of the axioms of group theory. How do we know this fact? We \nneed simply exhibit two models, two groups, one of which is abelian and the other not. \nFor these groups, I pick, oh say the [-47y cyclic group on 3 elements]\nand the [-497 symmetric group] $S_3$. The first is abelian, the second, not.\n\nIn order to reason about such models of set theory, one assumes the existence \nof "meta-sets" in some meta-theory. The entire "universe" of set theory is then a certain "meta-set" \nbehaving in a certain way. In case this feels too much like cheating,\n[-http://math.stackexchange.com/questions/531516/meta-theory-when-studying-set-theory this \nStackExchange answer] should help clear things up. In particular, the following quote from\nVanLiere's thesis:\n\n"\nSince these questions all have to do with first-order provability, we could take as our metatheory some very weak theory (such as Peano arithmetic) which is sufficient for formalizing first-order logic. However, as is customary in treatises about set theory, we take as our metatheory ZF plus the Axiom of choice in order to have at our disposal the infinitary tools of model theory. We will also use locutions such as ... which are only really justifiable in some even stronger metatheory with the understanding that they could be eliminated through the use of Boolean-valued models or some other device.\n"\n\nIn other words, it is possible to form these models in some weaker theory on which we have "more of a grip". The entirity\nof set theory are then special objects satisfying the axioms of this weaker theory. Is it possible to repeat this process\nad infinitum? No. But if we want, we could even deal with just a finite fragment of set theory: We assume that any mathematics\nwe want to do only needs a finite number of the (infinitely many) axioms of set theory. We then prove what we want about this \nfinite fragment. But we may as well have proved it about the whole theory. \n\nNow, pick (or construct) two specific objects of the meta-theory such\n that in one of them, the axiom of choice is true, and that\nin the other, the axiom of choice is false.\n\nTo obtain these two models requires vastly different approaches which will not be \ndescribed in detail here. \nMore detail can be found online\n in [-http://vanilla47.com/PDFs/Cryptography/Mathematics/Set%20Theory%20PDFs/SET%20THEORY.pdf \nKunen's text]. The consistency of choice is the easier direction, and involves constructing\nsomething called [-constructible_universe Gödel's constructible universe of sets] (or just\nGödel's universe or the constructible universe). The consistency of the negation of choice\nis more difficult, and involves a technique developed by \n[-https://en.wikipedia.org/wiki/Paul_Cohen_(mathematician) Paul Cohen]\ncalled [-forcing_mathematics forcing].\n\n#A Rose by Any Other Name: Alternative Characterizations of AC#\nThere are a few similar ways to state the axiom of choice. For example:\n\nGiven any set $C$ containing non-empty sets which are pairwise [-disjoint_set disjoint]\n(any two sets in $C$ do not [-5sb intersect]), \nthen there is some set $X$ that intersects each of the sets in $C$ in exactly one element.\n\nThere are also many alternative theorems which at first glance appear to be very different from the \naxiom of choice, but which are actually equivalent to it, in the sense that each of the theorems\ncan both be proven from the axiom of choice and be used to prove it (in conjunction\nwith the other ZF axioms).\n\nA few examples:\n - Zorn's Lemma (Described in more detail below).\n - Well-Ordering Theorem (Also described in more detail below).\n - The [-5zs product] of non-empty sets is non-empty.\n - Tarski's Theorem for Binary [-product_mathematics Products] \n (from the quote at the start of this article) that $A$ is [-bijection bijective] to $A \\times A$. \n - Every [-4bg surjective function] has a [-inverse_of_function right inverse].\n - Given two sets, either \n - Every [-3w0 vector space] has a [-vector_space_basis basis]\n - Every [-connected_graph connected graph] has a [-spanning_tree spanning tree].\n - For any pair of sets have comparable[-4w5 cardinality]: for any pair of sets\n$X$ and $Y$, either they are the same size, or one is smaller than the other.\n \nMore examples can be found on the [-https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents Wikipedia page].\n\nBecause of the intuitive nature of some of these statements (especially that products are non-empty,\nthat vector spaces have bases and that cardinalities are comparable), they are often used as evidence\nfor the truth, or motivation for the use, of the Axiom of Choice.\n\n#Zorn's Lemma? I hardly Know her!#\nThe following is a specific example of a very common way in which the axiom of choice is used, called Zorn's Lemma. \nIt is a statement equivalent to the axiom of choice, but easier to use in many mathematical applications.\n\nThe statement is as follows:\n\nEvery [-3rb partially ordered set] (poset) for which every [-chain_order_theory chain] has an [-upper_bound_mathematics upper bound]\nhas a [-maximal_mathematics maximal element]. \n\nIn other words, if you have an ordered set $X$ and you consider any linearly ordered subset $C$, and there is some \nelement $u \\in X$ which is at least as large as any element in $C$, i.e., $u \\geq c$ for any $c \\in C$, then there\nis an element $m \\in X$ which is maximal, in the sense that it is at least as big as any comparable \nelement of $X$, i.e., for any $x \\in X$, it holds that $m \\not< x$. \n(It is maximal, but not necessarily a global maximum. There is nothing in $X$\nlying above $m$, but there may be incomparable elements).\n\nAgain, this is provable for a finite poset X, and for some infinite posets X, but not provable in general. \n\nNow, why is this rather arcane statement useful? \n\nWell, often for some type of mathematical structure we are interested in, \nall the structures of that type form a poset under inclusion, and if a maximal\nsuch structure existed, it would have a particularly nice property.\nFurthermore, for many structures, a union of all the structures in a \nupwards-increasing chain of such structures (under inclusion) is itself\na structure of the right type, as well as an upper-bound for the chain.\nThen Zorn's lemma gives us the maximal structure we are looking for.\n\nAs an example, consider a (esp. infinite-dimensional) vector space $V$, and trying to find a basis\nfor $V$.\n\nChoose any element $v_1 \\in V$. Now consider all possible [-linear_independence linearly independent] sets\ncontaining $v$. These form a poset (which contains at least the set $\\{v_1\\}$ since that is linearly independent).\nNow consider any chain, possibly infinite, of such sets. It looks like\n $\\{v_1\\} \\subseteq \\{v, v_2\\} \\subseteq \\{v_1, v_2, v_3 \\} \\subseteq \\cdots$. Then take the \n[-union union] of all the sets in the chain\n$ \\{v_1\\} \\cup \\{v_1, v_2\\} \\cup \\{v_1, v_2, v_3 \\} \\cdots = \\{v_1, v_2, v_3, \\ldots \\}$. Call it $B$.\nThen $B$ \ncontains all the elements in any of the sets in the chain. \n\nIt can be shown to be linearly independent, since if some element $v_i$ could be formed\nas a linear combination of finitely many other elements in $B$, then this could be done\nalready in one of the sets in the chain.\n\nThen every chain of such linearly indpendent sets has an upper bound, so\nthe hypothesis of Zorn's Lemma holds. Then by Zorn's Lemma, there is a maximal element\n$M$. By definition, this maximal element has no superset that is itself linearly independent.\nThis set of vectors also spans $V$ (every element of $V$ can be written as a linear\ncombination of vectors in $M$), since if it did not, then there would be a vector\n$v \\in V$ which is linearly independent of $M$ (cannot be written as a linear \ncombination of vectors in $M$) and then the set $M \\cup \\{v\\}$, which is $M$ adjoined\nwith $v$, strictly contains $M$, contradicting the maximality of $M$. \n\nSince the definition of a basis is a maximal linearly independent set spanning $V$,\n the proof is done.\nQED.\n\n\nOne might wonder why Zorn's lemma is even necessary. Why could we not just have picked the \nunion of the chain as our basis? In a sense, we could have, provided we use the correct chain.\nFor example, the chain of two elements $\\{v_1\\}$ and $\\{v_1, v_2\\}$ is not sufficient. We need\nan infinite chain (and, in fact, a [-transfinite large enough infinite] chain at that) . \nBut there is a difference between being able to \nprove that any chain has an upper bound, and being able to actually choose a specific chain that works.\nIn some sense, without Zorns lemma, we can reason in a very general vague way that, yes, the chains all\nhave upper bounds, and there *might* be a long enough chain, and if there is then its upper bound will\nbe the maximal element we need. Zorn's lemma formalizes this intuition, and without it,\nlemma we can't always pin down a specific chain which works.\n\nNote how Zorn's Lemma allows us to make infinitely many arbitrary choices as far as selecting\nelements of the infinite basis for the vector space is concerned. In general, this is where \nZorn's lemma comes in useful. \n\nThe upper bounds are necessary. For example, in the [-45h natural numbers] $\\mathbb{N}$, the \nentirety of the natural numbers forms a chain. This chain has no upper bound in $\\mathbb{N}$. Also,\nthe natural numbers do not have a maximal element.\n\nNote that it may also be possible for a maximal element to exist without there being an upper bound.\nConsider for example the natural numbers with an 'extra' element which is not comparable to any of the\nnumbers. Then this is a perfectly acceptable poset. Since this extra element is incomparable, then in \nparticular there is no \n\n#Getting Your Ducks in a Row, or, Rather, Getting Your Real Numbers in a Row: The Well-Ordering Principle#\nA [540 linearly ordered set] is called well-ordered if any of its non-empty subsets has a least element.\n\nFor example, the [-45h natural numbers] $\\mathbb{N}$ are well-ordered. Consider any non-empty subset of the natural \nnumbers (e.g., $\\{42, 48, 64, \\ldots\\}$. It has a least element (e.g., $42$). \n\nThe positive [-4bc real numbers] (and in fact the positive [-4zq rational numbers]) \nare not well-ordered. \nThere is no least element, since for any number bigger than zero (e.g. 1/3) it is possible to find a smaller number\n(e.g. 1/4) which is also bigger than zero.\n\nThe well-ordering principle states that any set is [-bijection bijective] to some well-ordered one. \nThis basically states that you can have a well-ordered set of any size. \n\nTo see why this is surprising, try imaging a different linear order on the reals such that any subset\nyou may choose - *Any* subset - has a least element. \n\nAgain, the Axiom of Choice allows us to do this.\n\nIn fact if we are always able to well order sets, then\nwe are able to use it to make choice functions: imagine you needed to choose an element from each set\nin a set of sets, then you can just choose the least element from each set. \n\n#AC On a Budget: Weaker Versions of the Axiom#\nThere are also theorems which do not follow from ZF, and which do follow from AC, but are not strong\nenough to use to prove AC. What this is equivalent to saying is that there are models of set theory\nin which these theorems are true, but for which the axiom of choice does not hold in full generality.\n\nA few examples of such theorems:\n\n - The Hausdorff paradox and Banach-Tarski paradox (mentioned above).\n - A [-union_mathematics union] of [-countble_infinity countably many] countable sets is countable.\n - The [-axiom_of_dependent_choice axiom of dependent choice] (given a non-empty set $X$ and\n([-entire_relation entire]) [-binary_relation binary relation] $R$, there exists a sequence \n$(x_n)_{n \\in \\mathbb{N}}$ such that $x_n$ is $R$-related to $x_{n+1}$) .\n - The [-axiom_of_countable_choice axiom of countable choice] (every countable set of sets \nhas a choice function).\n - Every [-field_mathematics field] has an [-algebraic_closure algebraic closure]\n - Existence of non-principal [-ultrafilter ultrafilters].\n - Gödel's [-completeness_theorem completeness theorem] for first-order logic.\n - [-booelan_prime_ideal_theorem Boolean Prime Ideal Theorem] (useful for proving existence \nof non-principal ultrafilters and Gödel's completenes theorem).\n - The [-excluded_middle Law of excluded middle] for logic.\n\nMore examples may be found on \n[-https://en.wikipedia.org/wiki/Axiom_of_choice#Results_requiring_AC_.28or_weaker_forms.29_but_weaker_than_it\nthe Wikipedia page].\n\n#And In Related News: The Continuum Hypothesis#\nIntuitively, the [-continuum_hypothesis Continuum Hypothesis] (CH) states that there is no\nset strictly bigger than the set of all [-45h natural numbers], \nbut strictly smaller than the set of all [-4bc real numbers]. (These are two\n[-infnity infinite] sets, but they are [-2w0 different infinities]). \n\nThe formal statement concerns [-4w5 cardinality] of sets. In particular, \nit states that there is no set which has cardinality strictly larger than the\nset of natural numbers, but strictly smaller than the set of real numbers.\n\nIt is called the 'Continuum Hypothesis' because it concerns the size of the\ncontinuum (the set of real numbers) and was hypothesized to be true by Georg\nCantor in 1878. \n\nIt is again independent of the Zermelo-Fraenkel axioms, and this was proven in the same manner and\nat the same time as the proof of the independence of AC from ZF (described in more detail above).\n\nIn fact, the continuum hypothesis was shown to be independent even from ZFC, (ZF with the Axiom of Choice).\nHowever, the continuum hypothesis *implies* the Axiom of Choice under ZF. In other words, given the \nZF axioms, if you know that the Axiom of Choice is true then you do not yet know anything about the truth\nof Continuum Hypothesis. However, if you know that the Continuum Hypothesis is true, then you know the \nAxiom of Choice must also be true.\n\nThe [-generalized_continuum_hypthesis Generalized Continuum Hypthosis] (GCH) is, well, a generalized\nversion of the Continuum Hypothesis, stating that not only are there no sets of size lying strictly\nbetween the natural numbers $\\mathbb{N}$ and the reals $\\mathbb{R}$, \nbut that for any set $X$, there is no set of size\nlying strictly between the sizes of $X$ and the [-6gl power set] $P(X)$. In particular, note\nthat the reals $\\mathbb{R}$ is of the same size as the power set $P(\\mathbb{N})$ of the naturals,\nso that GCH implies CH. It is also strictly stronger than CH (it is not implied by CH).\n\n#Axiom of Choice Considered Harmful: Constructive Mathematics and the Potential Pitfalls of AC#\nWhy does the axiom of choice have such a bad reputation with [-constructive_mathematics constructive] \nmathematicians? \n\nIt is important to realise that some of the reasons mathematicians had for doubting AC are no longer \nrelevant. Because of some of its counter-intuitive results, mathematicians \n\nTo understand this view, it is necessary to understand more about the constructive view in general. %TODO\n\n(Posting a list of links here until I have a better idea of what to write, in approx. reverse order of relevance / usefulness)\nSee also \n\n\n - [-https://en.wikipedia.org/wiki/Axiom_of_choice#Criticism_and_acceptance this section of the Wikipedia page on AC]\n - [-http://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice this StackOverflow question],\n - [-http://comjnl.oxfordjournals.org/content/49/3/345.full this paper by Per Martin-Löf on the history and problems with the Axiom of Choice]\n - [-https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ this post by Greg Muller on why the Axiom of Choice is wrong]\n - [-http://www.iep.utm.edu/con-math/ this post on constructive mathematics on the Internet Encyclopaedia of Philosophy]\n - [-http://scienceblogs.com/goodmath/2007/05/27/the-axiom-of-choice/ this post \non Good Math Bad Math by Mark C. Chu-Carroll],\n - [-https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/ this\npost by Terrance Tao on the usefulness of the axiom in "concrete" mathematics], \n - [-http://www.math.canterbury.ac.nz/php/groups/cm/faq/ this post regarding constructive mathematics on the University of Canterbury website]\n - [-https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=8&ved=0ahUKEwiPnovngczPAhVM1hoKHflFDtMQFghYMAc&url=http%3A%2F%2Fmath.fau.edu%2Frichman%2FDocs%2FIntrview.tex&usg=AFQjCNFJZO1QGPzG1Cqe_8XcnlzMCrlvsA&sig2=R2goExeTv4EPJ6TF9ks5Yw\nthis "interview with a constructivist"]\n - [-http://plato.stanford.edu/entries/mathematics-constructive/ this page on Standford Enclyclopaedia on constructive mathematics]\n - [-http://plato.stanford.edu/entries/intuitionism/ this page on Stanford Encyclopaedia on intuitionistic mathematics]\n\n#Choosing Not to Choose: Set-Theoretic Axioms Which Contradict Choice#\nIt is also possible to assume axioms which contradict the axiom of choice. For example, there is the \n[-axiom_of_determinancy Axiom of Determinancy]. This axiom states that for any two-person [-game_mathematics game] \nof a certain type, one player has a winning strategy. \n%TODO\n\n#I Want to Play a Game: Counterintuitive Strategies Using AC#\nThere are a few examples of very counter-intuitive solutions to (seemingly) impossible challanges\nwhich work only in the presence of the Axiom of Choice.\n\nExamples may be found in the following places:\n\n - [-https://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/ Countably many prisoners have to guess hats with no information transfer, yet all but a finite number may go free.]\n - [-http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice Countably many people wear hats with \nreal numbers. They can only see everyone else's hats. Shouting at the same time, all but finitely many are guaranteed to guess correctly.]\n - [-http://math.stackexchange.com/questions/613506/real-guessing-puzzle Countably many mathematicians go into countably many identical rooms each containing countably many boxes containing real numbers. Each mathematician opens all but one box, and guesses the number in the unopened box. 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