Implications

We have the following indirect implication of form equivalence classes:

172 \(\Rightarrow\) 182
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 \(\Rightarrow\) 104 clear
104 \(\Rightarrow\) 182 clear

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.

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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