Editing Axiom of choice (section)

==Criticism and acceptance==
A proof requiring the axiom of choice may establish the existence of an object without explicitly [[definable set|defining]] the object in the language of set theory. For example, while the axiom of choice implies that there is a [[well-ordering]] of the real numbers, there are models of set theory with the axiom of choice in which no individual well-ordering of the reals is definable. Similarly, although a subset of the real numbers that is not [[Lebesgue measure|Lebesgue measurable]] can be proved to exist using the axiom of choice, it is [[consistent]] that no such set is definable.{{sfn|Fraenkel|Bar-Hillel|Lévy|1973|pp=69–70}}

The axiom of choice asserts the existence of these intangibles (objects that are proved to exist, but which cannot be explicitly constructed), which may conflict with some philosophical principles.{{sfn|Rosenbloom|2005|page=147}} Because there is no [[Canonical form|canonical]] well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired (as is often the case in [[category theory]]). This has been used as an argument against the use of the axiom of choice.

Another argument against the axiom of choice is that it implies the existence of objects that may seem counterintuitive.<ref>{{harvnb|Dawson|2006}}: "The axiom of choice, though it had been employed unconsciously in many arguments in analysis, became controversial once made explicit, not only because of its non-constructive character, but because it implied such extremely unintuitive consequences as the Banach–Tarski paradox."</ref> One example is the [[Banach–Tarski paradox]], which says that it is possible to decompose the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original. The pieces in this decomposition, constructed using the axiom of choice, are [[non-measurable set]]s.

Despite these seemingly paradoxical results, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics. But the debate is interesting enough that it is considered notable when a theorem in ZFC (ZF plus AC) is [[logical equivalence|logically equivalent]] (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type that requires the axiom of choice to be true.

Theorems of ZF hold true in any [[model theory|model]] of that theory, regardless of the truth or falsity of the axiom of choice in that particular model. The implications of choice below, including weaker versions of the axiom itself, are listed because they are not theorems of ZF. The Banach–Tarski paradox, for example, is neither provable nor disprovable from ZF alone: it is impossible to construct the required decomposition of the unit ball in ZF, but also impossible to prove there is no such decomposition. Such statements can be rephrased as conditional statements—for example, "If AC holds, then the decomposition in the Banach–Tarski paradox exists." Such conditional statements are provable in ZF when the original statements are provable from ZF and the axiom of choice.