Axiom Of Choice

We have the following indirect implication of form equivalence classes:

214 \(\Rightarrow\) 82 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
214 \(\Rightarrow\) 9	clear
9 \(\Rightarrow\) 82	clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
214:	\(Z(\omega)\): For every family \(A\) of infinite sets, there is a function \(f\) such that for all \(y\in A\), \(f(y)\) is a non-empty subset of \(y\) and \(\|f(y)\|=\aleph_{0}\).
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
82:	\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

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