Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
292 \(\Rightarrow\) 90 The axiom of choice and linearly ordered sets, Howard, P. 1977, Fund. 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 \(\Rightarrow\) 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
292:

\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).

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.

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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