Implications

We have the following indirect implication of form equivalence classes:

79 \(\Rightarrow\) 306
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
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
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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