Implications

Hypothesis: HR 307:

If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it Vitali equivalence classes} are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in {\Bbb Q})(x-y=q)\).

Conclusion: HR 224:

There is a partition of the real line into \(\aleph_1\) Borel sets \(\{B_\alpha: \alpha<\aleph_1\}\) such that for some \(\beta <\aleph_1\), \(\forall\alpha <\aleph_1\), \(B_{\alpha}\in G_{\beta}\). (\(G_\beta\) for \(\beta < \aleph_1\) is defined by induction, \(G_0=\{A: A\) is an open subset of \({\Bbb R}\}\) and for \(\beta > 0\),

  • \(G_\beta =\left\{\bigcup^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists\xi <\beta)(A_i\in G_\xi)\,\right\}\) if \(\beta\) is even and
  • \(G_\beta = \left\{\bigcap^\infty_{i=0}A_{i}: (\forall i\in\omega) (\exists \xi < \beta)(A_{i}\in G_\xi)\,\right\}\) if \(\beta\) is odd.)

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

Name Statement
\(\cal M5(\aleph)\) Solovay's Model An inaccessible cardinal \(\aleph\) is collapsed to \(\aleph_1\) in the outer model and then \(\cal M5(\aleph)\) is the smallest model containing the ordinals and \(\Bbb R\)

Code: 3

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