Axiom of choice

Stated more formally:

Another formulation of the axiom of choice (AC) states:

Until the late 19th century, the axiom of choice was often used implicitly. For example, a proof might have, after establishing that the set S contains only non-empty sets, said "let F(X) be one of the members of X for all X in S." Here, the existence of the function F depends on the axiom of choice.

The axiom might seem at first glance to be obviously true and unobjectionable: if there are several boxes, each containing at least one item, the axiom simply states that one can choose exactly one item from each box. The existence of a choice function is indeed straightforward and uncontroversial when only finite sets are concerned. In fact its existence can be proven from the other axioms of set theory, without the axiom of choice. More generally, the axiom of choice is not necessary for the existence of a choice function when one can come up with a rule to choose items from the sets. However, it is necessary when such a rule cannot be found, and applicable even when such a rule can be proven not to exist. Asserting the existence of a choice function in such cases is controversial. The controversy involves what it means to choose something from these sets, and what it means for a set to exist.

To see the issue, let us look at some sample sets.

1. Let X be any finite collection of non-empty sets.
:: Then f can be stated explicitly (out of set A choose a, ...), since the number of sets is finite.
:: Here the axiom of choice is not needed; the existence of the choice set follows from the other axioms of set theory.
:2. Let X be the collection of all non-empty subsets of the natural numbers {0, 1, 2, 3, ... }.
:: Then f can be the function that chooses the smallest element in each set.
:: Again the axiom of choice is not needed, since we have a rule for doing the choosing.
:3. Let X be the set of all sub-intervals of (0,1) with a length greater than 0.
:: Then f can be the function that chooses the midpoint of each interval.
:: Again the axiom of choice is not needed.
:4. Let X be the collection of all non-empty subsets of the reals.
:: Now the existence of f is not so straightforward. There is no obvious definition of f that will guarantee success, and there are reasons to believe that such an f may not be definable. We cannot simply have f pick the smallest element as we did in example 2 because a set of real numbers need not have a smallest element; there is not, for example, a smallest rational number or a smallest positive real number. Perhaps under some ordering of the reals other than the usual one there would always be a smallest element. However, the other axioms of ZF set theory do not guarantee the existence of a well-ordering of the real numbers (or of any other uncountable set). In fact the statement that every set can be well-ordered is equivalent to the axiom of choice.

Theorems whose proofs involve the axiom of choice are always nonconstructive: they demonstrate the existence of something without telling us how to get it.

The axiom of choice has been proven to be logically independent of the remaining axioms of set theory; that is, it can be neither proven nor disproven from them (unless those remaining axioms contain an unknown contradiction). This is the result of work by Kurt Gödel and Paul Cohen.
Thus no contradictions arise if the axiom of choice is rejected. However, most mathematicians accept either it, or a weakened variant of it, because it makes their jobs easier. Despite this, there is some study of systems in which the axiom of choice is either not true or at least not assumed (see also axiom of regularity). It is important to be aware of which proofs in mathematics use the axiom of choice and which do not. Furthermore, by contrast, there is a school of mathematical philosophy known as constructivism which asserts that proofs that assert the existence of something without defining how to get it are invalid, and this school rejects the axiom of choice.

The truth or falsity of the axiom of choice does not appear to be relevant to the physical world. The reason appears to be that all known sets corresponding to physical objects appear to be finite or at most countable, and with this limitation a choice function can always be defined, using the principle of induction, rendering the axiom of choice superfluous. Or, one could argue that all physically measurable quantities behave well under approximation and hence countable sets are adequate for mathematical modelling in the real world.

One reason that some mathematicians dislike the axiom of choice is that it implies the existence of some bizarre counter-intuitive objects. An example of this is the Banach-Tarski paradox which says in effect that it is possible to "carve up" the 3-dimensional solid unit ball into finitely many pieces and, using only rotation and translation, reassemble the pieces into two balls each with the same volume as the original. Note that the proof, like all proofs involving the axiom of choice, is an existence proof only: it does not tell us how to carve up the unit sphere to make this happen, it simply tells us that it can be done.

One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up. There are also a remarkable number of important statements that are equivalent to the axiom of choice, most important among them Zorn's lemma and the well-ordering theorem: every set can be well-ordered. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle.

Several central theorems in different branches of mathematics require the axiom of choice (or a weak version of it, such as the Boolean prime ideal theorem, the axiom of countable choice, or the axiom of dependent choice):