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

tidy | en | SemBr

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

Table of Contents

Countable Principle of Choice

Relationship to other axioms

Strengthenings

Graph View

Backlinks

Table of Contents

Countable Principle of Choice

Relationship to other axioms

Strengthenings

Footnotes

Graph View

Backlinks