Incomparable $\omega_1$-like models of set theory

TL;DR

This paper demonstrates that certain embedding theorems for countable models of set theory do not extend to uncountable $oldsymbol{ extomega_1}$-like models, revealing a rich diversity and incomparability among these models under specific hypotheses.

Contribution

It shows the failure of Hamkins embedding analogues for uncountable models, constructing many pairwise incomparable $oldsymbol{ extomega_1}$-like models with various embedding properties.

Findings

Existence of $2^{oldsymbol{ extomega_1}}$ many pairwise incomparable $oldsymbol{ extomega_1}$-like models of ZFC.

A transitive $oldsymbol{ extomega_1}$-like model of ZFC that does not embed into its constructible universe.

An $oldsymbol{ extomega_1}$-like model of PA with a hereditarily finite set structure not universal for such models.

Abstract

We show that the analogues of the Hamkins embedding theorems, proved for the countable models of set theory, do not hold when extended to the uncountable realm of $\omega_1$-like models of set theory. Specifically, under the $\diamondsuit$ hypothesis and suitable consistency assumptions, we show that there is a family of $2^{\omega_1}$ many $\omega_1$-like models of ZFC, all with the same ordinals, that are pairwise incomparable under embeddability; there can be a transitive $\omega_1$-like model of ZFC that does not embed into its own constructible universe; and there can be an $\omega_1$-like model of PA whose structure of hereditarily finite sets is not universal for the $\omega_1$-like models of set theory.