the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty...

abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel...

In mathematics, the axiom of dependent choice, denoted by D C {\displaystyle {\mathsf {DC}}} , is a weak form of the axiom of choice ( A C {\displaystyle...

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty...

theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally...

foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role...

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...

satisfy the axiom of dependent choice but not the full axiom of choice, the knowledge that a particular proof only requires dependent choice can be useful...

lemma Axiom of global choice Axiom of countable choice Axiom of dependent choice Boolean prime ideal theorem Axiom of uniformization Axiom of real determinacy...

set (with cardinality ℵ0) of positive integers. If the axiom of countable choice (a weaker version of the axiom of choice) holds, then ℵ0 is smaller...

equivalent to the axiom of choice (often called AC, see also Axiom of choice § Equivalents). Ernst Zermelo introduced the axiom of choice as an "unobjectionable...

L(R) of a set theory, which accepts only a weak form of the axiom of choice (AC) but contains all real and all ordinal numbers. Some consequences of AD...

form of the axiom of choice"—namely, the axiom of global choice: There exists a global choice function G {\displaystyle G} defined on the class of all...

that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed:...

Axiom of Choice is a southern California based world music group of Iranian émigrés who perform a modernized fusion style rooted in Persian classical...

If the axiom of choice is true, this transfinite sequence includes every cardinal number. If the axiom of choice is not true (see Axiom of choice § Independence)...

theory, Luce's choice axiom, formulated by R. Duncan Luce (1959), states that the relative odds of selecting one item over another from a pool of many items...

proof of BCT1 for arbitrary complete metric spaces requires some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to the axiom of dependent...

and f({2, 7}) = 2 is a choice function on X. Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering...

model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF...

up axiom of choice in Wiktionary, the free dictionary. Axiom of choice is an axiom of set theory. Axiom of choice may also refer to: Axiom of Choice (band)...

universal nets was given in 1992 by Paul Chernoff. All of the above proofs use the axiom of choice (AC) in some way. For instance, the third proof uses...

specialization of Tychonoff's theorem that requires only the ultrafilter lemma (and not the full strength of the axiom of choice) states that any product of compact...

cardinality of the real numbers." In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers:...

implies the axioms of replacement, separation, union, and global choice. It is equivalent to the combination of replacement, union, and global choice in Von...

properties). Generalizations of this axiom are explored in inner model theory. The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel...

(Zermelo–Fraenkel axioms without the axiom of choice) alone. The axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove...

constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory. Classical measure theory...

ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum...

The axiom is not a theorem of ZFC (assuming ZFC is consistent), but unlike the full axiom of determinacy (AD), which contradicts the axiom of choice, it...