Virtual Set Theory
: Taking the Blue Pill

Student thesis: Doctoral ThesisDoctor of Philosophy (PhD)

Abstract

The first part of this thesis is an analysis of the virtual large cardinals, being critical points of set-sized generic elementary embeddings where the target model is a subset of the ground model. We show that virtually measurables are equiconsistent with virtually strongs, and that virtually Woodins are virtually Vopěnka. We separate most of these large cardinals, but show that such separations do not hold within core models. We define prestrong cardinals, being an equivalent characterisation of strongs, but which in a virtual setting are strictly weaker than virtually strongs. We show that the existence of this separation is equivalent to the existence of virtually rank-into-rank cardinals in the universe, and that virtually Berkeley cardinals can be characterised in the same fashion with On being virtually pre-Woodin but not virtually Woodin, answering a question by Gitman and Hamkins. Building on the work of Wilson, we show that the virtual version of the Weak Vopěnka Principle is equivalent to a weakening of virtually pre-Woodins. We end the first part with several indestructibility results, including that a slight strengthening of the virtually supercompacts is always indestructible by <κ-directed closed forcings. The second part is concerned with connections between the virtual large cardinals and other set-theoretic objects. We analyse cardinals arising from a certain filter game, for various lengths of the game. When the games are finite we show that this results in a characterisation of the completely ineffable cardinals, and at length ω we arrive at another characterisation of the virtually measurable cardinals. At length ω + 1 the cardinals become equiconsistent with a measurable cardinal, and at uncountable cofinalities the cardinals are downward absolute to K below 0 ¶ . The results in this section answer most of the open questions raised in [Holy and Schlicht, 2018]. We also introduce the notion of ideal-absolute properties of forcings, being properties such that generic elementary embeddings can be characterised by ideals in the ground model. We show that several properties are ideal-absolute, which includes an improvement of an unpublished theorem of Foreman. This also results in another characterisation of completely ineffables.
Date of Award29 Sep 2020
Original languageEnglish
Awarding Institution
  • The University of Bristol
SupervisorViveka Erlandsson (Supervisor) & Philip D Welch (Supervisor)

Keywords

  • Set theory
  • Virtual large cardinal
  • Large cardinal
  • Inner model theory
  • Generic large cardinal
  • Precipitous ideal
  • Ramsey-like cardinal
  • Filter game
  • Ineffable cardinal
  • Forcing
  • Indestructibility

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