Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
41 \(\Rightarrow\) 9 clear
9 \(\Rightarrow\) 57 clear

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

Howard-Rubin Number Statement
41:

\(W_{\aleph _{1}}\): For every cardinal \(m\), \(m \le \aleph_{1}\) or \(\aleph_{1}\le m \).

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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