We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
100 \(\Rightarrow\) 9 |
On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc. The Axiom of Choice, Jech, 1973b, page 162 problem 11.8 |
9 \(\Rightarrow\) 82 | clear |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
---|---|
100: | Weak Partition Principle: For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\). |
9: | Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite. |
82: | \(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.) |
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