Implications

We have the following indirect implication of form equivalence classes:

21 \(\Rightarrow\) 104
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
21 \(\Rightarrow\) 23 Zermelo's Axiom of Choice, Moore, [1982]
23 \(\Rightarrow\) 25 Über dichte Ordnungstypen, Hausdorff, F. 1907, Jber. Deutsch. Math.
25 \(\Rightarrow\) 34 clear
34 \(\Rightarrow\) 104 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
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 \(|\bigcup A| = \aleph _{\alpha }\).

25:

\(\aleph _{\beta +1}\) is regular for all ordinals \(\beta\).

34:

\(\aleph_{1}\) is regular.

104:

There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

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