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Despite its usefulness, many mathematicians reject the Axiom of Choice. This rejection is based on the belief that all [[mathematical proofs]] should be constructive. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Suppose we consider a set consisting of many pairs of shoes. The Axiom of Choice says that we can create a new set by selecting just the left shoe from each pair. It is obvious that we can do this for shoes, since left and right shoes are distinguishable. What if the pairs of objects in question are <i>not</i> distinguishable and it is not acceptable to say 'just pick any one of them'?

Despite its usefulness, many mathematicians reject the Axiom of Choice. This rejection is based on the belief that all [[mathematical proofs]] should be constructive. AC, by its very nature, is nonconstructive, since it merely asserts that a choice function exists, but does not give an explicit method for its construction. Suppose we consider a set consisting of many pairs of shoes. The Axiom of Choice says that we can create a new set by selecting just the left shoe from each pair. It is obvious that we can do this for shoes, since left and right shoes are distinguishable. What if the pairs of objects in question are <i>not</i> distinguishable and it is not acceptable to say 'just pick any one of them'?

Put another way, it is objectionable to insist that a function exists based on the Axiom of Choice, and then to derive results based on the existence of that function, when in fact there is no known example of such function and no algorithm for identifying it. This objection to use of the Axiom of Choice is analogous to the objection to the use of non-[[falsifiable]] claims in science.  It strains logic to insist that something is true when it cannot be verified, and if it were false no one would even know.

Put another way, it may be objectionable to insist that a function exists based on the Axiom of Choice, and then to derive results based on the existence of that function, when in fact there may be no known way to construct such a function. This objection to use of the Axiom of Choice is analogous to the objection to the use of non-[[falsifiable]] claims in science.  Since the Axiom of Choice is independent of the other axioms, results provable with it may be verified with the other axioms.

Use of the Axiom of Choice has led to some ostensibly absurd results.  In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identitical size, thereby creating 2 out of only 1.  This paradox is "proven" only through use of the Axiom of Choice, and the authors of this "proof" did so to criticize this Axiom.  One attempt to resolve this apparent contradiction is to say that physical spheres are not really infinitely divisible, and that the creation of 2 out of 1 results from this assumption of infinitely divisible parts.<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

Use of the Axiom of Choice has led to some seemingly absurd results.  In the [[Banach-Tarski Paradox]], the Axiom of Choice is used to prove that a solid sphere of infinitely divisible parts may be chopped up and reconstructed as two new spheres of identical size, thereby creating 2 out of only 1.  This paradox is proven only through use of the Axiom of Choice, and the authors of this proof did so to criticize this Axiom.  One attempt to resolve this apparent contradiction is to say that physical spheres are not Lebesgue measurable.<ref>http://www.kuro5hin.org/story/2003/5/23/134430/275</ref>

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.

The formulation of a constructive Axiom of Choice is one of three major problems which challenge 21st century logicians.