Implications

We have the following indirect implication of form equivalence classes:

90 \(\Rightarrow\) 34
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
90 \(\Rightarrow\) 51 Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
51 \(\Rightarrow\) 25 Choice and cofinal well-ordered subsets, Morris, D.B. 1969, Notices Amer. Math. Soc.
25 \(\Rightarrow\) 34 clear

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

Howard-Rubin Number Statement
90:

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

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 \(\beta\).

34:

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

Comment:

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