Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
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
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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