Countable Principle of Choice

The Countable Axiom of Choice $A C ℕ$ is a possible axiom of material set theory and rather weak choice principle:¹ #m/def/set

(\forall 𝔐 𝐵) (\forall 𝑃 \subseteq ℕ \times 𝐵) [(\forall 𝑛 \in ℕ) (\exists 𝑦 \in 𝐵) 𝑃 (𝑛, 𝑦) ⟹ (\exists 𝑓 : ℕ \to 𝐵) (\forall 𝑛 \in ℕ) 𝑃 (𝑛, 𝑓 (𝑛))]

which is to say, if $𝐵$ is a set and $𝑃 \subseteq ℕ \times 𝐵$ is a left-total Relation set, i.e. relates every $𝑛 \in ℕ$ with at least one $𝑏 \in 𝐵$ , then there exists a choice function that selects such a $𝑏$ for each $𝑛$ . Thus countable sequences of independent choices are always possible.

Relationship to other axioms

Strengthenings

Over $Z F$ , $A C ℕ$ is a strict weakening of the Axiom of Dependent Choice and thus the Axiom of Choice.

#state/tidy | #lang/en | #SemBr

2006. Notes on set theory, ¶8.12, p. 114 ↩

Countable Principle of Choice

Relationship to other axioms

Strengthenings

Footnotes