Material set theory

Axiom of Dependent Choice

The Axiom of Dependent Choice DC is a possible axiom of material set theory providing a sufficiently strong choice principle for most mathematics:¹ set

(∀𝔐⁡𝐴)(∀𝑃⊆𝐴×𝐴)[𝑎∈𝐴∧(∀𝑥∈𝐴)(∃𝑦∈𝐴)𝑃(𝑥,𝑦)⟹(∃𝑓:ℕ→𝐴)[𝑓(0)=𝑎∧(∀𝑛∈ℕ)𝑃(𝑓(𝑛),𝑓(𝑛+1))]]

which is to say it is possible to make a countable sequence of choices, each dependent on the last.

Relationship to other axioms

Strengthenings

Over ZF, DC is a strict weakening of the Axiom of Choice.

Weakenings

Over ZF, DC is a strengthening of the Countable Axiom of Choice, which only allows for independent sequences of choices.

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develop | en | SemBr

Footnotes

1. 2006. Notes on set theory, ¶8.13, p. 114 ↩