Implications

We have the following indirect implication of form equivalence classes:

23 \(\Rightarrow\) 34
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.
25 \(\Rightarrow\) 34 clear

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 \(\beta\).

34:

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

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