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Nontrivial automorphisms of $\mathcal P(ω)/\mathrm{Fin}$ in Cohen models

Will Brian, Alan Dow

Abstract

We show that if $κ< \aleph_ω$ Cohen reals are added to a model of $\mathsf{CH}$, then there are nontrivial automorphisms of $\mathcal P(ω)/\mathrm{Fin}$ in the extension. Under some further hypotheses on the ground model, namely the existence of long enough sage Davies trees (which follows from $\mathsf{SCH}$ plus $\square_λ$ for every $λ$ with $\mathrm{cf}(λ) = ω$), we prove the same result for cardinals $κ\geq \aleph_ω$ as well. This extends a result a Shelah and Steprāns, who proved the result for $κ= \aleph_2$.

Nontrivial automorphisms of $\mathcal P(ω)/\mathrm{Fin}$ in Cohen models

Abstract

We show that if Cohen reals are added to a model of , then there are nontrivial automorphisms of in the extension. Under some further hypotheses on the ground model, namely the existence of long enough sage Davies trees (which follows from plus for every with ), we prove the same result for cardinals as well. This extends a result a Shelah and Steprāns, who proved the result for .
Paper Structure (3 sections, 2 theorems, 9 equations)

This paper contains 3 sections, 2 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Sage Davies trees after Cohen forcing
  3. Building the automorphisms

Key Result

Theorem 2.1

Let $\kappa$ be an infinite cardinal with $\kappa^\omega = \kappa$, and suppose $\left\langle M_\alpha \colon \alpha < \kappa \right\rangle$ is a sage Davies tree for $\kappa$. If $G$ is $\mathrm{Fn}(\kappa,2)$-generic over $V$, then the sequence $\left\langle M_\alpha[G] \colon \alpha < \kappa \rig

Theorems & Definitions (7)

  • Theorem 2.1
  • proof
  • Claim
  • Claim
  • Theorem 3.1
  • proof
  • proof : Proof of part $(3)$ of the main theorem