In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.

In many cases such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin.

For example for any ( even infinite ) collection of pairs of shoes, one can pick out the left shoe from each pair to obtain an appropriate selection, but for an infinite collection of pairs of socks ( assumed to have no distinguishing features ), such a selection can be obtained only by invoking the axiom of choice.

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

Contemporary set theorists also study axioms that are not compatible with the axiom of choice, such as the axiom of determinacy.

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

Thus the negation of the axiom of choice states that there exists a set of nonempty sets which has no choice function.

The axiom of choice asserts the existence of such elements ; it is therefore equivalent to:

There are many other equivalent statements of the axiom of choice.

These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

With this alternate notion of choice function, the axiom of choice can be compactly stated as

The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

" In any of these systems, removal of the one axiom which is equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry.

Despite these facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics.

Together these results establish that the axiom of choice is logically independent of ZF.

Without the axiom of choice, these theorems may not hold for mathematical objects of large cardinality.

The much stronger axiom of determinacy, or AD, implies that every set of reals is Lebesgue measurable, has the property of Baire, and has the perfect set property ( all three of these results are refuted by AC itself ).

Categorical axiom systems for these structures can be obtained in stronger logics such as second-order logic.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent.

For example, many results about Borel sets are proved by transfinite induction on the ordinal rank of the set ; these ranks are already well-ordered, so the axiom of choice is not needed to well-order them.

The first three of these characterizations can be proven equivalent in Zermelo – Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles.

However, these may all be different if the axiom of choice fails.

Given the short distances of super special stages compared to the regular special stages and consequent near-identical times for the frontrunning cars, it is very rare for these spectator-oriented stages to decide rally results, though it is a well-known axiom that a team can't win the rally at the super special, but they can certainly lose it.

Examples of these axioms include the combination of Martin's axiom and the Open colouring axiom which, for example, prove that, while the continuum hypothesis implies the opposite.

Of course, one can easily find regular spaces that are not T < sub > 0 </ sub >, and thus not Hausdorff, such as an indiscrete space, but these examples provide more insight on the T < sub > 0 </ sub > axiom than on regularity.

In fact these uses of AC are essential: in 1950 Kelley proved that Tychonoff's theorem implies the axiom of choice.

Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1 / n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε.

Together these committees reflected a strong consensus to halt the uncontrolled sprawl of London and other large cities, under the axiom if we can build better, we can live better.

( Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a homomorphism of abelian groups from the K-theory K ( X ) into the rational cohomology of X.

Without this axiom, we simply have the axioms which define a totally ordered field, and there are many non-isomorphic models which satisfy these axioms.

In these theories, Zermelo's axiom of separation does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different way.

In reverse mathematics, one starts with a framework language and a base theory — a core axiom system — that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems.

For these reasons, such set theorists tend to consider large cardinal axioms to have a preferred status among extensions of ZFC, one not shared by axioms of less clear motivation ( such as Martin's axiom ) or others that they consider intuitively unlikely ( such as V = L ).

Another argument against the axiom of choice is that it implies the existence of counterintuitive objects.

This discovery led to the well-known axiom: like-charged objects repel and opposite-charged objects attract.

In mathematics, an axiom of countability is a property of certain mathematical objects ( usually in a category ) that requires the existence of a countable set with certain properties, while without it such sets might not exist.

" one behavioral axiom, ' the propensity to truck, barter, and exchange one thing for another ,' where the objects of trade I will interpret to include not only goods, but also gifts, assistance, and favors out of sympathy ... whether it is goods or favors that are exchanged, they bestow gains from trade that humans seek relentlessly in all social transactions.

By NBG's axiom schema of Class Comprehension, all objects satisfying any given formula in the first order language of NBG form a class ; if the class would not be a set in ZFC, it is an NBG proper class.

Note that in this case, the axiom of extensionality must be formulated to apply only to objects that are not urelements.

A conceptual model is a representation of some phenomenon, data or theory by logical and mathematical objects such as functions, relations, tables, stochastic processes, formulas, axiom systems, rules of inference etc.

The final axiom ( TR 4 ) is called the " octahedral axiom " because drawing all the objects and morphisms gives the skeleton of an octahedron, four of whose faces are distinguished triangles.