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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.

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 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>

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.