Implications

We have the following indirect implication of form equivalence classes:

51 \(\Rightarrow\) 25
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.

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 \(\beta\).

Comment:

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