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An ordinal-connection axiom as a weak form of global choice under the GCH

Abstract

The minimal ordinal-connection axiom \(MOC\) was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that \(MOC\) is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, \(MOC\) is in fact equivalent to the \({{\,\mathrm{GCH}\,}}\). Our main results then show that \(MOC\) corresponds to a weak version of global choice in models of the \({{\,\mathrm{GCH}\,}}\): it can fail in models of the \({{\,\mathrm{GCH}\,}}\) without global choice, but also global choice can fail in models of \(MOC\).

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Notes

  1. A rank function is slow if the collection of objects of rank \(\alpha \) only has size \(|\alpha |\) for each infinite \(\alpha \). This slowness condition is clearly not satisfied by the usual rank function with respect to the von Neumann hierarchy.

  2. If r(x) happens to be finite, we let \(r(x)^+=r(x)+1\).

  3. This will avoid having to notationally deal with an additional top level of our hierarchy in the case when \({{\,\mathrm{Ord}\,}}\cap K\) is a successor ordinal.

  4. Clearly, from (H1) we obtain \(|K_\alpha |\ge |\alpha |\), so that together with (H3), we obtain \(|K_\alpha |=|\alpha |\) for any infinite \(\alpha \).

  5. Note that Acceptability for L[A], which was used as assumption in [2,  Proposition 5.1], implies that for every regular uncountable cardinal \(\kappa \), we have \(L[A]_\kappa =H(\kappa )^{L[A]}\).

  6. Recall that for any name \(\sigma \) and automorphism \(\pi \), the name \(\pi (\sigma )\) is obtained by recursively applying \(\pi \) to all conditions appearing in (the transitive closure of) the name \(\sigma \).

  7. Note that none of the below properties refers to forcing statements, but to properties of the actual name \(\dot{K}_{\alpha +1}\) in the ground model V.

  8. This will imply that \(K_{\alpha +1}\) is an \(\alpha \)-size subset of \(H(\alpha ^+)\) in \({\mathbb {P}}\)-generic extensions.

  9. This will imply that \(\bigcup _{\beta <\alpha ^+}K_\beta =H(\alpha ^+)\) in \({\mathbb {P}}\)-generic extensions.

References

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  4. Hamkins, J.D.: (https://mathoverflow.net/users/1946/joel-david hamkins). Does ZFC prove the universe is linearly orderable? An answer on MathOverflow. https://mathoverflow.net/q/110823 (version: 2012-11-03)

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Correspondence to Peter Holy.

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The research of the first author was supported by CNPq and Fapesp. The research of the second author was supported by the Italian PRIN 2017 Grant Mathematical Logic: models, sets, computability. The authors would like to thank the anonymous referee of the paper for a number of helpful comments. The authors have no competing interests to declare that are relevant to the content of this article.

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Freire, R.A., Holy, P. An ordinal-connection axiom as a weak form of global choice under the GCH. Arch. Math. Logic (2022). https://doi.org/10.1007/s00153-022-00838-2

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Keywords

  • Ordinals
  • Ordinal-Connection Axioms
  • Generalized Continuum Hypothesis
  • Global choice

Mathematics Subject Classification

  • 03E10
  • 03E65
  • 03E50
  • 03E25
  • 03E35
  • 03E45
  • 03E70