We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 359 \(\Rightarrow\) 20 | clear |
| 20 \(\Rightarrow\) 101 |
Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic |
| 101 \(\Rightarrow\) 100 | clear |
| 100 \(\Rightarrow\) 369 |
Communication sur les recherches de la th'eorie des ensembles, Lindenbaum, A. 1926, C. R. Soc. Sci. Lett. Varsovie |
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). |
| 101: | Partition Principle: If \(S\) is a partition of \(M\), then \(S \precsim M\). |
| 100: | Weak Partition Principle: For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\). |
| 369: | If \(\Bbb R\) is partitioned into two sets, at least one of them has cardinality \(2^{\aleph_0}\). |
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