axiom of countable choice

Noun

axiom of countable choice

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.
   • 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...2000, Bruno Poizat, translated by Moses Klein, A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, page 169
   • we have no way to infer
     ∃R∀n[P0n→∀x(R_nx→Fx)∧n=Nx:R_nx)]
     without an axiom of countable choice.2012, Richard G. Heck, Jr., Reading Frege's Grundgesetze, Oxford University Press, page 271, But, once again, while we can easily prove ∀n[P0n→∃G(∀x(Gx→Fx)∧n=Nx:Gx)]

   Synonyms:

   axiom of denumerable choice