Implications

We have the following indirect implication of form equivalence classes:

164 \(\Rightarrow\) 306
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 \(\Rightarrow\) 203 clear
203 \(\Rightarrow\) 306 clear

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.

203:

\(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function.

306:

The set of Vitali equivalence classes is linearly orderable. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow (\exists q\in{\Bbb Q})(x-y = q)\).).

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