Implications

We have the following indirect implication of form equivalence classes:

202 \(\Rightarrow\) 169
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
202 \(\Rightarrow\) 91 note-75
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 272 Models of set theory containing many perfect sets, Truss, J. K. 1974b, Ann. Math. Logic
272 \(\Rightarrow\) 169 clear

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

Howard-Rubin Number Statement
202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

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.

169:

There is an uncountable subset of \({\Bbb R}\) without a perfect subset.

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