Implications

We have the following indirect implication of form equivalence classes:

51 \(\Rightarrow\) 58
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
51 \(\Rightarrow\) 208 Choice and cofinal well-ordered subsets, Morris, D.B. 1969, Notices Amer. Math. Soc.
208 \(\Rightarrow\) 58 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.

208:

For all ordinals \(\alpha\), \(\aleph_{\alpha+1}\le 2^{\aleph_\alpha}\).

58:

There is an ordinal \(\alpha\) such that \(\aleph(2^{\aleph_{\alpha }})\neq\aleph_{\alpha +1}\). (\(\aleph(2^{\aleph_{\alpha}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph _{\alpha}}\).)
Mathias [1979], p 126.

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