Implications

We have the following indirect implication of form equivalence classes:

114 \(\Rightarrow\) 104
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
114 \(\Rightarrow\) 90 Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
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
34 \(\Rightarrow\) 104 clear

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

Howard-Rubin Number Statement
114:

Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.)

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.

104:

There is a regular uncountable aleph. Jech [1966b], p 165 prob 11.26.

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