Implications

We have the following indirect implication of form equivalence classes:

305 \(\Rightarrow\) 93
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
305 \(\Rightarrow\) 306 Cardinality of the set of Vitali equivalence classes, Kanovei, V.G. 1991, Mat. Zametki
306 \(\Rightarrow\) 93 L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.

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

Howard-Rubin Number Statement
305:

There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}.

306:

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

93:

There is a non-measurable subset of \({\Bbb R}\).

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