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

TL;DR

This paper demonstrates that adding Cohen reals to certain models of set theory creates nontrivial automorphisms of the Boolean algebra , extending previous results to larger cardinals under specific hypotheses.

Contribution

It extends the known existence of nontrivial automorphisms of  in Cohen models to larger cardinals using advanced set-theoretic hypotheses.

Findings

Nontrivial automorphisms exist after adding Cohen reals for _\u03c9 under CH.

Extension of results to __ for larger __ with additional hypotheses.

Results build on and generalize previous work by Shelah and Steprns.

Abstract

We show that if $\kappa < \aleph_\omega$ Cohen reals are added to a model of $\mathsf{CH}$, then there are nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ 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 $\square_\lambda$ for every $\lambda$ with $\mathrm{cf}(\lambda) = \omega$), we prove the same result for cardinals $\kappa \geq \aleph_\omega$ as well. This extends a result a Shelah and Stepr\={a}ns, who proved the result for $\kappa = \aleph_2$.