Axiom Of Choice

We have the following indirect implication of form equivalence classes:

112 \(\Rightarrow\) 108 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
112 \(\Rightarrow\) 90	Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 79
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 \(\Rightarrow\) 38	The Axiom of Choice, Jech, [1973b]
38 \(\Rightarrow\) 108	clear

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

Howard-Rubin Number	Statement
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
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.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.
108:	There is an ordinal \(\alpha\) such that \(2^{\aleph _{\alpha}}\) is not the union of a denumerable set of denumerable sets.

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