Destructibility and Axiomatizability of Kaufmann Models

TL;DR

This paper investigates the set-theoretic properties of Kaufmann models, focusing on their destructibility via forcing and axiomatizability in extended logics, revealing independence results from ZFC.

Contribution

It demonstrates the independence of Kaufmann models' destructibility and axiomatizability from ZFC, connecting these issues to Aronszajn trees and advanced set-theoretic techniques.

Findings

Destructibility of Kaufmann models is independent of ZFC.

Axiomatizability of Kaufmann models in L_{ω_1, ω}(Q) is independent of ZFC.

Results relate Kaufmann models to incompactness phenomena like Aronszajn trees.

Abstract

A Kaufmann model is an $\omega_1$-like, recursively saturated, rather classless model of $\mathrm{PA}$ or $\mathrm{ZF}$. Such models were constructed by Kaufmann under the combinatorial principle $\diamondsuit_{\omega_1}$ and Shelah showed they exist in $\mathrm{ZFC}$ by an absoluteness argument. Kaufmann models are an important witness to the incompactness of $\omega_1$ similar to Aronszajn trees. In this paper we look at some set theoretic issues related to this motivated by the seemingly na\"{i}ve question of whether such a model can be "killed" by forcing without collapsing $\omega_1$. We show that the answer to this question is independent of $\mathrm{ZFC}$ and closely related to similar questions about Aronszajn trees. As an application of these methods we also show that it is independent of $\mathrm{ZFC}$ whether or not Kaufmann models can be axiomatized in the logic $L_{\omega_1, \omega} (Q)$ where $Q$ is the quantifier "there exists uncountably many".