We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 359 \(\Rightarrow\) 20 | clear |
| 20 \(\Rightarrow\) 22 | |
| 22 \(\Rightarrow\) 26 | clear |
| 26 \(\Rightarrow\) 209 | note-72 |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 359: | If \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \( |A_{x}| \le |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| \le |\bigcup_{x\in S} B_{x}|\). |
| 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). |
| 22: | \(UT(2^{\aleph_{0}},2^{\aleph_{0}},2^{\aleph_{0}})\): If every member of an infinite set of cardinality \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\), then the union has power \(2^{\aleph_{0}}\). |
| 26: | \(UT(\aleph_{0},2^{\aleph_{0}},2^{\aleph_{0}})\): The union of denumerably many sets each of power \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\). |
| 209: | There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets. |
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