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 \(\Rightarrow\) 40 |
On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys. |
| 40 \(\Rightarrow\) 86-alpha | clear |
| 86-alpha \(\Rightarrow\) 196-alpha |
Successive large cardinals, Bull Jr., E. L. 1978, Ann. Math. Logic |
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 \(S \precsim M\). |
| 40: | \(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325. |
| 86-alpha: | \(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(|X| = \aleph_{\alpha }\), then \(X\) has a choice function. |
| 196-alpha: | \(\aleph_{\alpha}\) and \(\aleph_{\alpha+1}\) are not both measurable. |
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