Dependent choice, properness, and generic absoluteness

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Abstract

We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to -preserving symmetric submodels of forcing extensions. Hence, not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in and. Our results confirm as a natural foundation for a significant portion of classical mathematics and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

Original languageEnglish
Pages (from-to)225–249
Number of pages25
JournalThe Review of Symbolic Logic
Volume14
Issue number1
Early online date2 Jul 2020
DOIs
Publication statusPublished - Jul 2020

Keywords

  • Axiom of Choice
  • PFA
  • The Chang model
  • forcing axioms
  • generic absoluteness
  • proper forcing
  • the Chang model

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