Implications

We have the following indirect implication of form equivalence classes:

164 \(\Rightarrow\) 389
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
164 \(\Rightarrow\) 91 Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 139
139 \(\Rightarrow\) 389

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

Howard-Rubin Number Statement
164:

Every non-well-orderable set has an infinite subset with a Dedekind finite power set.

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.

139:

Using the discrete topology on 2, \(2^{\cal P(\omega)}\) is compact.

389:

\(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function.  \ac{Keremedis} \cite{1999b}.

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