axiom of countable choice

Noun

 * 1)  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.
 * 2) * 2012, Richard G. Heck, Jr., Reading Frege's Grundgesetze,, page 271,
 * But, once again, while we can easily prove
 * $$\forall n[P^{*=}0n\rightarrow\exists G(\forall x(Gx\rightarrow Fx)\land n = Nx:Gx)]$$
 * we have no way to infer
 * $$\exists R\forall n[P^{*=}0n\rightarrow \forall x({R_n}x\rightarrow Fx)\land n = Nx:{R_n}x)]$$
 * without an axiom of countable choice.
 * 1) * 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, 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.
 * 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.

Translations

 * Finnish: numeroituva valinta-aksiooma