Implications

We have the following indirect implication of form equivalence classes:

91 \(\Rightarrow\) 272
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 272 Models of set theory containing many perfect sets, Truss, J. K. 1974b, Ann. Math. Logic

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

Howard-Rubin Number Statement
91:

\(PW\):  The power set of a well ordered set can be well ordered.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

272:

There is an \(X\subseteq{\Bbb R}\) such that neither \(X\) nor \(\Bbb R - X\) has a perfect subset.

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