Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
23 \(\Rightarrow\) 207-alpha clear

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

Howard-Rubin Number Statement
23:

\((\forall \alpha)(UT(\aleph_{\alpha},\aleph_{\alpha}, \aleph_{\alpha}))\): For every ordinal \(\alpha\), if \(A\) and every member of \(A\) has cardinality \(\aleph_{\alpha}\), then \(|\bigcup A| = \aleph _{\alpha }\).

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 \(2^{\aleph_{\alpha}}\).

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