Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
172 \(\Rightarrow\) 34	On hereditarily countable sets, Jech, T. 1982, J. Symbolic Logic
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
172:	For every infinite set \(S\), if \(S\) is hereditarily countable (that is, every \(y\in TC(S)\) is countable) then \(\|TC(S)\|= \aleph_{0}\).
34:	\(\aleph_{1}\) is regular.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.

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