Implications

We have the following indirect implication of form equivalence classes:

54 \(\Rightarrow\) 54
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
54 \(\Rightarrow\) 54 A survey of recent results in set theory, Mathias, A.R.D. 1979, Period. Math. Hungar.

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

Howard-Rubin Number Statement
54:

For all infinite cardinals \(m\), \(m\) adj \(2^{m}\) implies \(m\) is an aleph.  
Note that \(m\) adj \(n\) iff \(m < n\) and \(\neg (\exists p) (m < p < n).\)
Mathias [1979] prob 1337, thm 1338 and prob 1339.

54:

For all infinite cardinals \(m\), \(m\) adj \(2^{m}\) implies \(m\) is an aleph.  
Note that \(m\) adj \(n\) iff \(m < n\) and \(\neg (\exists p) (m < p < n).\)
Mathias [1979] prob 1337, thm 1338 and prob 1339.

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