axiom of countable choice

Definitions

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.

Examples

“The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable. One example is the axiom of countable choice, which states that if A#95;0,A#95;1,#92;dotsA#95;n#92;dots form⟳ a denumerable set⟳ of nonempty sets, their product is nonempty.[…]The axiom of countable choice is constantly used in analysis; it is often hidden so as not to sow⟳ confusion in the minds of the students (who are inclined to accept⟳ anything desired) or of the professors (who do not like⟳ to shake⟳ the foundations of the discipline).”

“we have⟳ no way to infer ∃R∀n[P0n→∀x(R_nx→Fx)∧n=Nx:R_nx)] 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, LNCS 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.”

CEFR level