What is set theory with an axiom of choice?

Question

I get the idea of set theory however I do not understand the axiom of choice nor why it is significant.

Answer 1

The axiom of choice says that the Cartesian product of nonempty sets is nonempty. That is (for example), if you have sets A_1, A_2, A_3, A_4, ..., and each set contains at least one element, then the set A_1 x A_2 x A_3 x ... contains at least one element.



Actually, that's not quite what the axiom of choice says, but what I've given is an equivalent statement that's perhaps slightly easier to understand.



The axiom of choice is not a theorem; it is an assertion that is (usually) part of the definition of what is meant by the word "set."



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Within set theory, one has a list of axioms which are ways in which we think that sets are supposed to behave. One of the most popular set theories is Zermelo-Fraenkel Set Theory (which I will call ZF from now on). Some examples of axioms in this theory are



* If two sets have the same elements, then they are equal.

* If a and b are two sets, then there exists a set whose elements are a and b.

* There exists an infinite set.



When a mathematician uses the word "set," the mathematician means "an object which satisfies all the axioms of set theory." Usually, the axiom of choice is included as one of these axioms. (The axiom of choice is, then, just a reasonable assertion about how sets should behave, and is part of the definition of what is meant by the word "set.")



For historical reasons which I'll address later on, the axiom of choice is not usually considered one of the axioms of ZF; it's considered a special extra axiom (but still one that's part of the definition of what a "set" is).



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If you want to, it is possible to define a "set" to be something that satisfies all the axioms of ZF, but doesn't have to satisfy the axiom of choice. If you choose to define a set this way, then there will exist sets A_1, A_2, A_3, ... which all contain at least one element, but the Cartesian product A_1 x A_2 x A_3 x ... is the empty set. Most mathematicians feel that this is a stupid way to define "set," because it seems to go against most people's intuition about the way collections of objects ought to behave.



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So why is the axiom of choice not considered part of ZF, but is instead usually considered a special extra axiom? The main reason is that, historically, mathematicians realized that if the axiom of choice is used as part of the definition of "set," then it is possible to prove some things that might seem somewhat counter-intuitive. Some examples include



* The Well-Ordering Theorem: This says that any set can be well-ordered; that is, any set can be arranged in an order (from "smallest" to "largest") so that any nonempty subset has a smallest element.



Note that the natural numbers are well-ordered; any subset of the natural numbers has a smallest element. On the other hand, the real numbers are *not* well-ordered using the usual ordering, because the set {x | x > 0} does not have a smallest element in the usual ordering.



The Well-Ordering Theorem implies that it is possible to re-order the real numbers so that *every* subset has a smallest element (like what happens in the natural numbers). Historically, some mathematicians found this very counter-intuitive, and it caused them to feel that perhaps the axiom of choice was not a reasonable thing to include as part of the definition of a set. To make matters worse, this well-ordering is, in a sense, *non-constructive*--the axiom of choice can be used to prove that it exists, but it's (in some sense) not possible to fully describe all at once what this well-ordering actually is.



* The Banach-Tarski Paradox: This says that it is possible to take a sphere, cut it into finitely many pieces (sets), rotate and move those pieces (sets) finitely many times, and end up with two complete spheres of the exact same dimensions as the original.



To most people, the Banach-Tarski Paradox seems extremely counter-intuitive--but if one accepts the axiom of choice as part of the definition of "set," then it is possible to prove that it's true.



Most mathematicians today accept the Banach-Tarski Paradox as an accurate description of how mathematics works (because one must accept it as true if one accepts the axioms of ZF and the axiom of choice as part of the definition of "set").



Note that the "pieces" in the Banach-Tarski paradox are not contiguous--in fact, they are sets so complicated that referring to the "surface area" of such a set or the "volume inside" such a set doesn't make any sense.



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Nowadays, most mathematicians accept the axiom of choice as part of the definition of what a "set" is, and mathematicians who don't accept the axiom of choice are generally viewed as eccentric.



However, since a few mathematicians don't, it is sometimes considered good to specifically highlight when one uses the axiom of choice in a proof--then those mathematicians who don't accept it can be aware that, to them, the proof is not valid, because when I use the word "set" I mean something that satisfies the axiom of choice, and when they use the word "set," they mean something else entirely.

Answer 2

The axiom of choice is essentially the assumption that the Cartesian product of arbitrarily many non-empty sets is non-empty.



To motivate this, the Cartesian product of two non-empty sets A and B is defined to be the set that is denoted by A x B and constructed as { (a,b) : a in A and b in B }; that is, A x B is the set of tuples such that the first member is in A and the second is in B.



Now let us suppose that we have an index set I_n = {1,2,...,n} containing the positive integers up to n; suppose further we have n corresponding sets S_i for i in I_n, all of which are non-empty. We define the n-fold Cartesian product S_1 x S_2 x ... x S_n to be the set of all n-tuples such that the i-th element is drawn from S_i; that is, the Cartesian product is given by { (s_1,s_2,...,s_n) : s_i in S_i }.



We still have no problem. But we can make this more general. Let us no longer assume that the indexing set I_n consists of n positive integers. Instead, replace I_n by I, a completely arbitrary indexing set that may not even be countable. Suppose for each object i in I there is a corresponding non-empty set S_i. We want to define the Cartesian product of all the S_i.



Because of the general nature of I, we can no longer do this like we did before. We therefore have to do something a bit odd. Let U denote the union of S_i over all i in I. We define the desired Cartesian product as the set of all mappings



s : I -> U



satisfying



s(i) in S_i



for i in I.



Each mapping is effectively a choice of an element from S_i for each i, thus reproducing our previous definition in the case of I = {1,2,...,n} for n a positive integer.



The axiom of choice is essentially the assumption that at least one such mapping exists. It's called the axiom of choice because these mappings are often called choice mappings.



The axiom of choice first rose to prominence when it was seen to be independent of the other axioms of ZF set theory. That is, introducing the axiom of choice is guaranteed not to introduce an inconsistency, but at the same time the axiom cannot be proven in any way from the other axioms--it's logically independent. People will sometimes tell you otherwise, but in the end mathematicians today still don't quite know how to handle this. Most simply ignore it! It is good practice, however, to recognize when you use the axiom and to make explicit mention of it.

Answer 3

If you throw out the axiom of choice, what remains can neither prove the axiom nor its negation. It is a strictly weaker system and many accepted things in mathematics can no longer be proven. For example you can no longer prove that every vector space has a basis, you can no longer prove that given any two sets either they are the same size or one is smaller [trichotomy for cardinal numbers]. Some proofs of things that are possible without the axiom of choice are much easier to prove with the axiom of choice.



Not just set theory, but any part of mathematics that uses the infinite, is significantly changed without the axiom of choice, including analysis. Most mathematicians accept it because without it life is just too difficult.



On balance, accepting the axiom of choice has some undesirable consequences, such as the following: it is possible to "chop up a sphere" into a finite number of "pieces" and reassemble them into a sphere with twice the volume, and no holes or gaps in it. If this makes you roll your eyes, consider rejecting the axiom of choice.

Answer 4

The axiom choice needed for many existence theorem, for example for showing that every vector space has a basis.



Axiom of choice says that if we have infinitely many family of sets we can choose one element from each of this set.