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On a Question of Hamkins and Löwe on the modal logic of collapse forcing

Mohammad Golshani, William Mitchell

Abstract

Hamkins and Löwe asked whether there can be a model $N$ of set theory with the property that $N\equiv N[g]$ whenever $g$ is a generic collapse of a cardinal of $N$ onto $ω$. We give equiconsistency results for two weaker versions of this property. We also include a proof of Woodin's result that the consistency of the full Hamkins-Löwe property follows from that of a Woodin cardinal with an inaccessible above.

On a Question of Hamkins and Löwe on the modal logic of collapse forcing

Abstract

Hamkins and Löwe asked whether there can be a model of set theory with the property that whenever is a generic collapse of a cardinal of onto . We give equiconsistency results for two weaker versions of this property. We also include a proof of Woodin's result that the consistency of the full Hamkins-Löwe property follows from that of a Woodin cardinal with an inaccessible above.

Paper Structure

This paper contains 5 sections, 8 theorems, 19 equations.

Table of Contents

  1. introduction
  2. Upper bounds for the weak Hamkins-Löwe properties
  3. The strength of a positive answer
  4. Woodin's upper bound for the Hamkins-Löwe property
  5. Questions

Key Result

Theorem 1.1

If there is a Woodin cardinal with a inaccessible cardinal above, then there is a countable well founded model satisfying the Hamkins-Löwe property.

Theorems & Definitions (23)

  • Theorem 1.1: Woodin
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1: ultrafilter sequence
  • Definition 2.2: Radin forcing
  • proof : Proof of Theorem \ref{['thm:main']}(2), right to left
  • proof : Proof of Theorem \ref{['thm:main']}(1), right to left
  • Theorem 3.1: Philip Welch
  • proof
  • ...and 13 more