Implications

We have the following indirect implication of form equivalence classes:

51 \(\Rightarrow\) 315
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
51 \(\Rightarrow\) 25 Choice and cofinal well-ordered subsets, Morris, D.B. 1969, Notices Amer. Math. Soc.
25 \(\Rightarrow\) 315 clear

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

Howard-Rubin Number Statement
51:

Cofinality Principle: Every linear ordering has a cofinal sub well ordering.  Sierpi\'nski [1918], p 117.

25:

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

315:

\(\Omega = \omega_1\), where
\(\Omega = \{\alpha\in\hbox{ On}: (\forall\beta\le\alpha)(\beta=0 \vee (\exists\gamma)(\beta=\gamma+1) \vee\)
there is a sequence \(\langle\gamma_n: n\in\omega\rangle\) such that for each \(n\),
\(\gamma_n<\beta\hbox{ and } \beta=\bigcup_{n<\omega}\gamma_n.)\} \)

Comment:

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