Pointwise definable models of set theory

@article{Hamkins2013PointwiseDM,
  title={Pointwise definable models of set theory},
  author={Joel David Hamkins and David Linetsky and Jonas Reitz},
  journal={The Journal of Symbolic Logic},
  year={2013},
  volume={78},
  pages={139 - 156}
}
Abstract A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is… 

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THE SET-THEORETIC MULTIVERSE

  • J. Hamkins
  • Philosophy
    The Review of Symbolic Logic
  • 2012
It is argued that the continuum hypothesis is settled on the multiverse view by the authors' extensive knowledge about how it behaves in theMultiverse, and as a result it can no longer be settled in the manner formerly hoped for.

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