Set Theory/The axiom of choice

Definition (axiom of countable finite choice):

The axiom of countable finite choice states that whenever $(S_{n})_{n\in \mathbb {N} }$ is a countable family of non-empty sets, then there exists a sequence $(x_{n})_{n\in \mathbb {N} }$ such that $\forall n\in \mathbb {N} :x_{n}\in S_{n}$ .

Exercises

Prove that Zorn's lemma is equivalent to Tukey's lemma which states that whenever $X$ is a set and ${\mathcal {F}}\subseteq 2^{X}$ has the property that $A\in {\mathcal {F}}$ if and only if $B\in {\mathcal {F}}$ for all finite sets $B\subseteq A$ , then for all $Y\subseteq X$ there exists a maximal $Z\subseteq X$ among all sets in ${\mathcal {F}}$ that contain $Y$ (where we consider ${\mathcal {F}}$ to be ordered by inclusion).

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