Implications

We have the following indirect implication of form equivalence classes:

328 \(\Rightarrow\) 83
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
328 \(\Rightarrow\) 126 clear
126 \(\Rightarrow\) 82 note-76
82 \(\Rightarrow\) 83 Definitions of finite, Howard, P. 1989, Fund. Math.

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

Howard-Rubin Number Statement
328:

\(MC(WO,\infty)\): For  every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)

126:

\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

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.)

83:

\(E(I,II)\) Howard/Yorke [1989]: \(T\)-finite is equivalent to finite.

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