We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 20 \(\Rightarrow\) 21 | clear |
| 21 \(\Rightarrow\) 23 | Zermelo's Axiom of Choice, Moore, [1982] |
| 23 \(\Rightarrow\) 25 |
Über dichte Ordnungstypen, Hausdorff, F. 1907, Jber. Deutsch. Math. |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 20: | If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in S\}\) are families of pairwise disjoint sets and \( |A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| = |\bigcup_{x\in S} B_{x}|\). Moore [1982] (1.4.12 and 1.7.8). |
| 21: | If \(S\) is well ordered, \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets, and \(|A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}|= |\bigcup_{x\in S} B_{x}|\). G\. |
| 23: | \((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\). |
| 25: | \(\aleph _{\beta +1}\) is regular for all ordinals \(\beta\). |
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