Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a non-zero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ...
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 sets must have a choice function. That is, given a function with domain (where denotes the set of natural numbers) such that is a non-empty set for every , there exists a function with domain such that for every .
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The axiomofcountablechoice or axiomof denumerable choice, denoted ACω, is an axiomof set theory that states that every countable collection of non-empty...
the axiomofchoice, abbreviated AC or AoC, is an axiomof set theory equivalent to the statement that a Cartesian product of a collection of non-empty...
assuming the axiomofcountablechoice, a set is countable if its cardinality (the number of elements of the set) is not greater than that of the natural...
lemma Axiomof global choiceAxiomofcountablechoiceAxiomof dependent choice Boolean prime ideal theorem Axiomof uniformization Axiomof real determinacy...
the set of all natural numbers. A set that is equinumerous to a proper subset of itself is called Dedekind-infinite. The axiomofcountablechoice (ACω)...
cardinality of a Dedekind-infinite set in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiomofcountable choice...
consisting of all ordinals smaller than ω 1 {\displaystyle \omega _{1}} . If the axiomofcountablechoice holds, every increasing ω-sequence of elements of [...
that ai+1 is an element of ai for all i. With the axiomof dependent choice (which is a weakened form of the axiomofchoice), this result can be reversed:...
theories, the axiomof global choice is a stronger variant of the axiomofchoice that applies to proper classes of sets as well as sets of sets. Informally...
well-order. The axiomofchoice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other...
of constructive set theories. Axiomof dependent choice D C {\displaystyle {\mathrm {DC} }} : Countablechoice is implied by the more general axiom of...
ultrafilter lemma: A countable union ofcountable sets is a countable set. The axiomofcountablechoice (ACC). The axiomof dependent choice (ADC). Under ZF...
_{1}} are countable (finite or denumerable). Assuming the axiomofchoice, the union of a countable set ofcountable sets is itself countable. So ℵ 1 {\displaystyle...
with additional semantics of the following countably infinitely many axioms added (these can be easily formalized as an axiom schema): ∃ x 1 : ∃ x 2 :...
In order to provide a model of probability, these elements must satisfy probability axioms. In the example of the throw of a standard die, The sample space...
f(\omega )\neq 0} for at most countably many ω ∈ Ω {\displaystyle \omega \in \Omega } . Herrlich, Horst (2006). The axiomofchoice. Lecture Notes in Mathematics...
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