Implications

We have the following indirect implication of form equivalence classes:

164 \(\Rightarrow\) 251
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 \(\Rightarrow\) 251 consequence of the axiom of choice, Ash, C. J. 1975, J. Austral. Math. Soc. Ser. A.

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.

251:

The additive groups \(({\Bbb R},+)\) and \(({\Bbb C},+)\) are  isomorphic.

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