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Power $Σ_1$ in Card with two Woodin cardinals

Farmer Schlutzenberg

Abstract

Väänänen and Welch asked in the paper "When cardinals determine the power set: inner models and Härtig quantifier logic" which large cardinals are consistent with the power set operation $x\mapsto P(x)$ being $Σ_1$-definable in the predicate Card of all cardinals. We show that, relative to large cardinals, this property is consistent with the existence of two Woodin cardinals.

Power $Σ_1$ in Card with two Woodin cardinals

Abstract

Väänänen and Welch asked in the paper "When cardinals determine the power set: inner models and Härtig quantifier logic" which large cardinals are consistent with the power set operation being -definable in the predicate Card of all cardinals. We show that, relative to large cardinals, this property is consistent with the existence of two Woodin cardinals.
Paper Structure (1 section, 1 theorem, 8 equations)

This paper contains 1 section, 1 theorem, 8 equations.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

Suppose that $M_2^\#$ exists and is $(0,\omega_1+1)$-iterable. Then there is a proper class transitive inner model which models ZFC + "There are 2 Woodin cardinals" + "power is $\Sigma_1^{\mathrm{Card}}$".

Theorems & Definitions (23)

  • Theorem 1
  • Remark 2
  • proof : Proof of Theorem \ref{['tm:y=P(x)_Sigma_1^Card']}
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Claim 3
  • proof
  • Claim 4
  • ...and 13 more