Implications

We have the following indirect implication of form equivalence classes:

245 \(\Rightarrow\) 108
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]
38 \(\Rightarrow\) 108 clear

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.

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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