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 \(\Rightarrow\) 27 | clear |
| 27 \(\Rightarrow\) 31 | clear |
| 31 \(\Rightarrow\) 419 |
Metric spaces and the axiom of choice, De-la-Cruz-Hall-Howard-Keremedis-Rubin-2002A[2002A], Math. Logic Quart. |
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 \(|\bigcup A| = \aleph _{\alpha }\). |
| 27: | \((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36. |
| 31: | \(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. |
| 419: | UT(\(\aleph_0\),cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is cuf if it is a countable union of finite sets.) |
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