Axiom of countable choice

English

Noun

1. Template:Lb A weaker form of the axiom of choice that states that every countable collection of nonempty sets must have a choice function; equivalently, the statement that the direct product of a countable collection of nonempty sets is nonempty.
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   • 2012, Richard G. Heck, Jr., Reading Frege's Grundgesetze, Template:W, page 271,

     But, once again, while we can easily prove

     $\mathrm{\forall }n[P^{*=}0n\to \mathrm{\exists }G(\mathrm{\forall }x(Gx\to Fx)\wedge n=Nx:Gx)]$

     we have no way to infer

     $\mathrm{\exists }R\mathrm{\forall }n[P^{*=}0n\to \mathrm{\forall }x(R_{n}x\to Fx)\wedge n=Nx:R_{n}x)]$

     without an axiom of countable choice.

   • 2013, Valentin Blot, Colin Riba, On Bar Recursion and Choice in a Classical Setting, Chung-chien Shan (editor), Programming Languages and Systems: 11th International Symposium, APLAS 2013, Proceedings, Springer, Template:W 8301, page 349,

     We show how Modified Bar-Recursion, a variant of Spector's Bar-Recursion due to Berger and Oliva can be used to realize the Axiom of Countable Choice in Parigot's Lambda-Mu-calculus, a direct-style language for the representation and evaluation of classical proofs.

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Translations

• Finnish: Template:T

See also

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Further reading

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