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 ".

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.

Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated.

Errett Bishop argued that the axiom of choice was constructively acceptable, saying

In mathematics, the axiom of regularity ( also known as the axiom of foundation ) is one of the axioms of Zermelo – Fraenkel set theory and was introduced by.

Thus Post in his 1936 was also discounting Kurt Gödel's suggestion to Church in 1934 – 5 that the thesis might be expressed as an axiom or set of axioms.

This story is an allegory ; the android was primitive scholasticism, which was broken by the Summa of St Thomas, the daring innovator who first substituted the absolute law of reason for arbitrary divinity, by formulating that axiom which we cannot repeat too often, since it comes from such a master: " A thing is not just because God wills it, but God wills it because it is just.

Paul Joseph Cohen ( April 2, 1934 — March 23, 2007 ) was an American mathematician best known for his proof of the independence of the continuum hypothesis and the axiom of choice from Zermelo – Fraenkel set theory, the most widely accepted axiomatization of set theory.

Until the 19th century, few doubted the truth of the postulate ; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms.

Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that ( since it would then prove its own consistency, which Gödel had shown was impossible ).

If there was an intuitive element, it was to be isolated and represented separately as an axiom: from there on, the proof was to be purely logical and without gaps.

This equality was always maintained with scrupulous care ; the Sorbon repeated as an axiom, Omnes nos sumus socii et aequales, and referred to the college as pauperem nostram Sorbonem.

In cryptography, Kerckhoffs's principle ( also called Kerckhoffs's Desiderata, Kerckhoffs's assumption, axiom, or law ) was stated by Auguste Kerckhoffs in the 19th century: A cryptosystem should be secure even if everything about the system, except the key, is public knowledge.

The completeness of sentential calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann ( 1924 ), von Neumann ( 1927 ) and Herbrand ( 1931 ).

The idea was that the government and the central bank would maintain rough full employment, so that neoclassical notions — centered on the axiom of the universality of scarcity — would apply.

It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo – Fraenkel set theory.

Carathéodory's version of the first law of thermodynamics was stated in an axiom which refrained from defining or mentioning temperature or quantity of heat transferred.

The assignment axiom states that after the assignment any predicate holds for the variable that was previously true for the right-hand side of the assignment.

He was able to develop most of classical calculus, while using neither the axiom of choice nor proof by contradiction, and avoiding Georg Cantor's infinite sets.

This was proven to be possible by Miklós Laczkovich in 1990 ; the decomposition makes heavy use of the axiom of choice and is therefore non-constructive.

Mathematics decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.

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

In probability theory, Luce's choice axiom, formulated by R. Duncan Luce ( 1959 ), states that the probability of selecting one item over another from a pool of many items is not affected by the presence or absence of other items in the pool.

") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

Because of independence, the decision whether to use of the axiom of choice ( or its negation ) in a proof cannot be made by appeal to other axioms of set theory.

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 ).

We will abbreviate " Zermelo-Fraenkel set theory plus the negation of the axiom of choice " by ZF ¬ C.

Conjecture is contrasted by hypothesis ( hence theory, axiom, principle ), which is a testable statement based on accepted grounds.

Was the notion of " effective calculability " to be ( i ) an " axiom or axioms " in an axiomatic system, or ( ii ) merely a definition that " identified " two or more propositions, or ( iii ) an empirical hypothesis to be verified by observation of natural events, or ( iv ) or just a proposal for the sake of argument ( i. e. a " thesis ").

Rather, he regarded the notion of " effective calculability " as merely a " working hypothesis " that might lead by inductive reasoning to a " natural law " rather than by " a definition or an axiom ".

* John D. Caputo attempts to explain deconstruction in a nutshell by stating that: " Whenever deconstruction finds a nutshell — a secure axiom or a pithy maxim — the very idea is to crack it open and disturb this tranquility.

Some axiomatic set theories assure that the empty set exists by including an axiom of empty set ; in other theories, its existence can be deduced.

This axiom is often omitted because a binary operation is closed by definition.

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.