Implications

We have the following indirect implication of form equivalence classes:

117 \(\Rightarrow\) 321
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
117 \(\Rightarrow\) 319 clear
319 \(\Rightarrow\) 320 note-20
320 \(\Rightarrow\) 321 clear

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

Howard-Rubin Number Statement
117:

If \(\kappa\) is a measurable cardinal, then \(\kappa\) is the \(\kappa\)th inaccessible cardinal.

319:

Measurable cardinals are inaccessible.

320:

No successor cardinal, \(\aleph_{\alpha+1}\), is measurable.

321:

There does not exist an ordinal \(\alpha\) such that \(\aleph_{\alpha}\) is weakly compact and \(\aleph_{\alpha+1}\) is measurable.

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