We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 29 \(\Rightarrow\) 27 |
Unions of well-ordered sets, Howard, P. 1994, J. Austral. Math. Soc. Ser. A. |
| 27 \(\Rightarrow\) 31 | clear |
| 31 \(\Rightarrow\) 32 |
L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat. |
| 32 \(\Rightarrow\) 10 | clear |
| 10 \(\Rightarrow\) 358 | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 29: | If \(|S| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) 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, G. [1982], p 324. |
| 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. |
| 32: | \(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function. |
| 10: | \(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function. |
| 358: | \(KW(\aleph_0,<\aleph_0)\), The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). |
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