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 \(\Rightarrow\) 64 |
The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math. clear |
| 64 \(\Rightarrow\) 127 |
Amorphe Potenzen kompakter Raume, Brunner, N. 1984b, Arch. Math. Logik Grundlagenforschung |
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. |
| 64: | \(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.) |
| 127: | An amorphous power of a compact \(T_2\) space, which as a set is well orderable, is well orderable. |
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