Implications

We have the following indirect implication of form equivalence classes:

207-alpha \(\Rightarrow\) 209
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
207-alpha \(\Rightarrow\) 209 note-72

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
207-alpha:

\(UT(\aleph_{\alpha },\aleph_{\alpha}, <2^{\aleph_{\alpha }})\): The union of \(\aleph_{\alpha}\) sets each of cardinality \(\aleph_{\alpha}\) has cardinality less than \(2^{\aleph_{\alpha}}\).

209:

There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets.

Comment:

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