Notes

Description:

Content: From Mathias [1979], p 124:
Theorem: (Solovay) \(Con(ZF) \rightarrow Con(ZF + 43 (DC(\omega))\) + Every endomorphism of the additive group of the real line is continuous.
Theorem: (Levy) \(Con(ZF) \rightarrow Con(ZF + cf(\omega_1) = \omega + {\Bbb R}\) is the denumerable union of denumerable sets + \(\neg\Sigma^1_3 -\) AC).
In Figura [1981] a summary of these results is given. In addition Figura generalizes Form 34 (\(\aleph_{1}\) is regular), Form 38 (\({\Bbb R}\) is not a countable union of countable sets) and Form 170 (\({\Bbb R}\) has an uncountable well orderable subset.) These generalizations for regular cardinals \(\kappa\), are respectively, (in the notation of Figura [1981]):

  • \(\neg CP(\kappa ): \kappa^{+}\) is regular.
  • \(\neg WOP(\kappa ): \cal P(\kappa )\) has a well ordered subset of power not \(\le\kappa\).
  • \(\neg DP(\kappa ): \cal P(\kappa )\) is not the union of \(\kappa\) sets of power \(\le\kappa\).

In summary, the results are

  1. \(DP(\kappa )\) implies \(WOP(\kappa )\) and \(DP(\kappa )\) implies \(CP(\kappa )\).
  2. Con(ZF\( + CP(\kappa ) + WOP(\kappa ) + DP(\kappa )\))
  3. Con(ZF\( + CP(\kappa ) + \neg WOP(\kappa ) + \neg DP(\kappa ))\)
  4. Con(ZF\( + CP(\kappa ) +WOP(\kappa ) + \neg DP(\kappa )\))
  5. Con(ZF\( + WOP(\kappa)+\neg CP(\kappa)+ \neg DP(\kappa)\))
  6. Con(ZF\( + \{ \kappa : DP(\kappa ) \}\) is a proper class)

Howard-Rubin number: 3

Type: Summary

Back