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

Will Brian, Alan Dow

TL;DR

The paper introduces the parametrized diamond $\diamondsuit(\mathsf{LP})$ to force the Laver property over a given inner model and to generate nontrivial automorphisms of $\mathcal{P}(\omega)/\mathrm{Fin}$ via bowties of functions. It establishes that $\diamondsuit(\mathsf{LP})$ holds in several CH-based forcing models (e.g., Laver, Mathias, Miller, Sacks, Silver) and shows how bowties yield explicit nontrivial automorphisms, including involutions. It also proves a limitation: in the Mathias model, every automorphism is somewhere trivial, thereby constraining the automorphisms that can be obtained from $\diamondsuit(\mathsf{LP})$. Overall, the work connects combinatorial guessing principles with structural results on automorphisms, highlighting how forcing extensions shape the landscape of nontrivial automorphisms.

Abstract

We introduce a new parametrized diamond principle denoted $\diamondsuit(\mathsf{LP})$. 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 $\aleph_1$. 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-$ω_2$ 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 $\mathcal P(ω)/\mathrm{Fin}$; 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 $\mathcal P(ω)/\mathrm{Fin}$ are somewhere trivial in the Mathias model. This puts a limitation on the kinds of automorphisms obtainable from $\diamondsuit(\mathsf{LP})$.

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

TL;DR

The paper introduces the parametrized diamond to force the Laver property over a given inner model and to generate nontrivial automorphisms of via bowties of functions. It establishes that holds in several CH-based forcing models (e.g., Laver, Mathias, Miller, Sacks, Silver) and shows how bowties yield explicit nontrivial automorphisms, including involutions. It also proves a limitation: in the Mathias model, every automorphism is somewhere trivial, thereby constraining the automorphisms that can be obtained from . Overall, the work connects combinatorial guessing principles with structural results on automorphisms, highlighting how forcing extensions shape the landscape of nontrivial automorphisms.

Abstract

We introduce a new parametrized diamond principle denoted . 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 -like guessing principle implying the corresponding invariant is . Our principle is a -like guessing principle implying the Laver property holds over a given inner model, such as the ground model in a forcing extension. We show holds in many familiar models of obtained by forcing, namely those obtained from a model of by a length- 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 . We also prove that if holds over an inner model of then there are nontrivial automorphisms of ; in fact we get particularly nice automorphisms extending nontrivial involutions built around -points in the ground model. Additionally, we show that, like the Sacks model, all automorphisms of are somewhere trivial in the Mathias model. This puts a limitation on the kinds of automorphisms obtainable from .
Paper Structure (4 sections, 13 theorems, 27 equations)

This paper contains 4 sections, 13 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Bowties of functions
  3. $\diamondsuit(\mathsf{LP})$
  4. Automorphisms in the Mathias model

Key Result

Theorem 2.1

If there is a bowtie of functions then there is a nontrivial automorphism $F$ of $\mathcal{P}(\omega)/\mathrm{Fin}$. Furthermore, there is a closed $P$-set in $\omega^*$ with character $\aleph_1$, namely $K = \bigcap_{\alpha < \omega_1}(\omega \setminus D_\alpha)^*$, such that $F$ is trivial on $A$

Theorems & Definitions (24)

  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • ...and 14 more