Implications

We have the following indirect implication of form equivalence classes:

245 \(\Rightarrow\) 182
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 \(\Rightarrow\) 104 clear
104 \(\Rightarrow\) 182 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.

104:

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

182:

There is an aleph whose cofinality is greater than \(\aleph_{0}\).

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