Hypothesis: HR 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}.
Conclusion: HR 236:
If \(V\) is a vector space with a basis and \(S\) is a linearly independent subset of \(V\) such that no proper extension of \(S\) is a basis for \(V\), then \(S\) is a basis for \(V\).
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N44\) Gross' model | \(A\) is a vector space over a finite field withbasis \(B = \bigcup_{i\in \omega} B_i\) where the \(B_i\) are pairwisedisjoint and \(|B_i| = 4\) for each \(i\in\omega\) |
Code: 3
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