We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 168 \(\Rightarrow\) 100 | clear |
| 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 |
| 82 \(\Rightarrow\) 387 |
"Dense orderings, partitions, and weak forms of choice", Gonzalez, C. 1995a, Fund. Math. |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 168: | Dual Cantor-Bernstein Theorem:\((\forall x) (\forall y)(|x| \le^*|y|\) and \(|y|\le^* |x|\) implies \(|x| = |y|)\) . |
| 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.) |
| 387: | DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering \(<\) on a set \(X\) is dense if \((\forall x, y\in X)(x \lt y \to (\exists z \in X)(x \lt z \lt y))\) and is non-trivial if \((\exists x,y\in X)(x \lt y)\)). |
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