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 103:
If \((P,<)\) is a linear ordering and \(|P| > \aleph_{1}\) then some initial segment of \(P\) is uncountable. Jech [1973b], p 164 prob 11.21.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N14\) Morris/Jech Model | \(A = \bigcup\{A_{\alpha}: \alpha <\omega_1\}\), where the \(A_{\alpha}\)'s are pairwise disjoint, each iscountably infinite, and each is ordered like the rationals; \(\cal G\) isthe group of all permutations on \(A\) that leave each \(A_{\alpha}\) fixedand preserve the ordering on each \(A_{\alpha}\); and \(S = \{B_{\gamma}:\gamma < \omega_1\}\), where \(B_{\gamma}= \bigcup\{A_{\alpha}: \alpha <\gamma\}\) |
Code: 3
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