Set Theory/The axiom of choice

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Definition (axiom of countable finite choice):

The axiom of countable finite choice states that whenever is a countable family of non-empty sets, then there exists a sequence such that .

Exercises[edit]

  1. Prove that Zorn's lemma is equivalent to Tukey's lemma which states that whenever is a set and has the property that if and only if for all finite sets , then for all there exists a maximal among all sets in that contain (where we consider to be ordered by inclusion).