Implications

We have the following indirect implication of form equivalence classes:

23 \(\Rightarrow\) 25
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
23 \(\Rightarrow\) 25 Über dichte Ordnungstypen, Hausdorff, F. 1907, Jber. Deutsch. Math.

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

Howard-Rubin Number Statement
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 \(\beta\).

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