Implications

We have the following indirect implication of form equivalence classes:

41 \(\Rightarrow\) 84
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
41 \(\Rightarrow\) 9 clear
9 \(\Rightarrow\) 84 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
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.

84:

\(E(II,III)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite  if and only if \(\cal P(x)\) is Dedekind finite).

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