We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
71-alpha \(\Rightarrow\) 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 |
---|---|
71-alpha: | \(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). Jech [1973b], page 119. |
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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