Implications

We have the following indirect implication of form equivalence classes:

214 \(\Rightarrow\) 57
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
214 \(\Rightarrow\) 9 clear
9 \(\Rightarrow\) 57 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.

57:

If \(x\) and \(y\) are Dedekind finite sets then either \(|x|\le |y|\) or \(|y|\le |x|\).
Mathias [1979], p 125.

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