Implications

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 183-alpha:

There are no \(\aleph_{\alpha}\) minimal  sets.  That is, there are no sets \(X\) such that

  1. \(|X|\) is incomparable with \(\aleph_{\alpha}\)
  2. \(\aleph_{\beta}<|X|\) for every \(\beta < \alpha \) and
  3. \(\forall Y\subseteq X, |Y|<\aleph_{\alpha}\) or \(|X-Y| <\aleph_{\alpha}\).

List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N27\) Hickman's Model II Let \(A\) be a set with cardinality\(\aleph_1\) such that \(A=\{(a_{\alpha},b_{\beta}) : \alpha < \omega, \beta< \omega_1\}\)

Code: 3

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