In this article and other discussions of the Axiom of Choice the following abbreviations are common:

* AC – the Axiom of Choice.

* ZF – Zermelo – Fraenkel set theory omitting the Axiom of Choice.

* ZFC – Zermelo – Fraenkel set theory, extended to include the Axiom of Choice.

* Axiom of determinacy, a set theory axiom inconsistent with the Axiom of Choice

This is also known as Zermelo's theorem and is equivalent to the Axiom of Choice.

Ernst Zermelo introduced the Axiom of Choice as an " unobjectionable logical principle " to prove the well-ordering theorem.

To see this ( without the Axiom of Choice ), fix a basis of open sets.

During this period he started supervising Ph. D. students, such as James Halpern ( Contributions to the Study of the Independence of the Axiom of Choice ) and Edgar Lopez-Escobar ( Infinitely Long Formulas with Countable Quantifier Degrees ).

In set theory, König's theorem ( named after the Hungarian mathematician Gyula Kőnig, who published under the name Julius König ) colloquially states that if the Axiom of Choice holds, I is a set, m < sub > i </ sub > and n < sub > i </ sub > are cardinal numbers for every i in I, and < math > m_i < n_i

In advanced mathematics, some functions exist because of an axiom, such as the Axiom of Choice.

) From there, one can prove ( with the Axiom of Choice ) that the least such cardinal must be inaccessible.

Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

It was introduced by Kurt Gödel in his 1938 paper " The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis ".

* NAV ’ s best contemporary world music album: Axiom of Choice ( band ).

For example, by the above, arithmetic with the Axiom of Choice is a Skolem theory.

* The following equivalence requires the Axiom of Choice:

In 1970, Solovay demonstrated that the existence of a non-measurable set for Lebesgue measure is not provable within the framework of Zermelo – Fraenkel set theory in the absence of the Axiom of Choice, by showing that ( assuming the consistency of an inaccessible cardinal ) there is a model of ZF, called Solovay's model, in which countable choice holds, every set is Lebesgue measurable and in which the full axiom of choice fails.

The Axiom of Choice is equivalent to a fundamental result of point-set topology, Tychonoff's theorem, and also to the conjunction of two fundamental results of functional analysis, the Banach – Alaoglu theorem and the Krein – Milman theorem.

* Axiom of Choice

* Scholarpedia article Luce's Choice Axiom

In fact, a stronger statement can be made: IST is a conservative extension of ZFC: any internal formula that can be proven within internal set theory can be proven in the Zermelo – Fraenkel axioms with the Axiom of Choice alone.

The axiom is independent of ZFC ( assuming that it is consistent with ZFC ), unlike the full axiom of determinacy ( AD ), which contradicts the Axiom of Choice.

Unless the Axiom of Choice is relaxed, free complete boolean algebras generated by a set do not exist ( unless the set is finite ).

Assuming the Axiom of choice, is regular for each α.

Axiom 1: If a property is positive, then its negation is not positive.

Axiom 2: Any property entailed by — i. e., strictly implied by — a positive property is positive

Axiom 3: The property of being God-like is positive

Axiom 4: If a property is positive, then it is necessarily positive

Axiom 5: Necessary existence is positive

Axiom 6: For any property P, if P is positive, then being necessarily P is positive.

: Axiom 1: It is possible to single out positive properties from among all properties.

: Axiom 2: For all properties A, either A is positive or " not A " is positive.

: Axiom 3: The property of " being God-like ", G is a positive property.

: Axiom 4: If a property A is positive, then it is so in every possible world.

: Axiom 5: Necessary existence is a positive property ( Pos ( NE )).

Here is a quotation from a paper by Jan Łukasiewicz, Remarks on Nicod's Axiom and on " Generalizing Deduction ", page 180.

then, seen as a statement about cardinal numbers, it is equivalent to the Axiom of choice.

In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T < sub > 4 </ sub >: every two disjoint closed sets of X have disjoint open neighborhoods.

This condition is known as Axiom T < sub > 3 </ sub >.

The Axiom Ensemble is a student directed and managed group dedicated to well-known 20th century works.