We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
71-alpha \(\Rightarrow\) 9 | clear |
9 \(\Rightarrow\) 13 | clear |
13 \(\Rightarrow\) 199(\(n\)) | clear |
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. |
13: | Every Dedekind finite subset of \({\Bbb R}\) is finite. |
199(\(n\)): | (For \(n\in\omega-\{0,1\}\)) If all \(\varSigma^{1}_{n}\), Dedekind finite subsets of \({}^{\omega }\omega\) are finite, then all \(\varPi^1_n\) Dedekind finite subsets of \({}^{\omega} \omega\) are finite. |
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