2026/01/27 by Will Brian, Alan Dow · 1 voice
Mathematics · #math.GN #math.LO
We introduce a new parametrized diamond principle denoted \diamondsuit(LP). This principle is akin to the parametrized diamonds of Moore, Hrušák, and Džamonja, each of which corresponds to some cardinal invariant of the continuum, and gives a \diamondsuit-like guessing principle implying the corresponding invariant is ℵ1. Our principle \diamondsuit(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(LP) holds in many familiar models of ZFC obtained by forcing, namely those obtained from a model of CH by a length-ω2 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 ℵ1. We also prove that if \diamondsuit(LP) holds over an inner model of CH then there are nontrivial automorphisms of \mathcal P(ω)/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(ω)/Fin are somewhere trivial in the Mathias model. This puts a limitation on the kinds of automorphisms obtainable from \diamondsuit(LP).