# Set Theory/The axiom of choice

**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

- 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).