A parametrized $\diamondsuit$ for the Laver property and nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$

Abstract

We introduce a new parametrized diamond principle denoted $\diamondsuit(\mathsf{LP})$. This principle is akin to the parametrized diamonds of Moore, Hru\v{s}\'ak, and D\v{z}amonja, each of which corresponds to some cardinal invariant of the continuum, and gives a $\diamondsuit$-like guessing principle implying the corresponding invariant is $\aleph_1$. Our principle $\diamondsuit(\mathsf{LP})$ is a $\diamondsuit$-like guessing principle implying the Laver property holds over a given inner model, such as the ground model in a forcing extension. We show $\diamondsuit(\mathsf{LP})$ holds in many familiar models of $\mathsf{ZFC}$ obtained by forcing, namely those obtained from a model of $\mathsf{CH}$ by a length-$\omega_2$ countable support iteration of proper Borel posets with the Laver property. This is true for essentially the same reason that the usual parametrized diamonds hold in similarly described forcing extensions where their corresponding cardinal invariant is $\aleph_1$. We also prove that if $\diamondsuit(\mathsf{LP})$ holds over an inner model of $\mathsf{CH}$ then there are nontrivial automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$; in fact we get particularly nice automorphisms extending nontrivial involutions built around $P$-points in the ground model. Additionally, we show that, like the Sacks model, all automorphisms of $\mathcal P(\omega)/\mathrm{Fin}$ are somewhere trivial in the Mathias model. This puts a limitation on the kinds of automorphisms obtainable from $\diamondsuit(\mathsf{LP})$.