Axiom Of Choice

We have the following indirect implication of form equivalence classes:

245 \(\Rightarrow\) 38 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
245 \(\Rightarrow\) 34	The monadic theory of \(\omega_1\), Litman, A. 1976, Israel J. Math.
34 \(\Rightarrow\) 38	The Axiom of Choice, Jech, [1973b]

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

Howard-Rubin Number	Statement
245:	There is a function \(f :\omega_1\rightarrow \omega^{\omega}_1\) such that for every \(\alpha\), \(0 < \alpha < \omega_1\), \(f(\alpha )\) is a function from \(\omega\) onto \(\alpha\).
34:	\(\aleph_{1}\) is regular.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.

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