Twins: non-isomorphic models forced to be isomorphic. Part I

TL;DR

This paper investigates conditions under which non-isomorphic models of a first-order complete theory become isomorphic after forcing extensions, exploring the interplay between model theory and set-theoretic forcing.

Contribution

It provides sufficient conditions for models to become isomorphic after forcing, especially for theories with the independence property, advancing the understanding of model isomorphism in forcing extensions.

Findings

Models far from isomorphic can become isomorphic after forcing.

For theories with the independence property, forcing with no new ω-sequences suffices.

Specific forcing notions like adding branches to trees can induce isomorphisms.

Abstract

For which (first-order complete, usually countable) $T$ do there exist non-isomorphic models of $T$ which become isomorphic after forcing with a forcing notion $\mathbb{P}$? Necessarily, $\mathbb{P}$ is non-trivial; i.e.~it adds some new set of ordinals. It is best if we also demand that it collapses no cardinal. It is better to demand on the one hand that the models are non-isomorphic, and even \emph{far} from each other (in a suitable sense), but on the other hand, $\mathcal{L}$-equivalent in some suitable logic $\mathcal{L}$. We give sufficient conditions: for theories with the independence property, we proved this when $\mathbb{P}$ adds no new $\omega$-sequence. We may prove it ``for some $\mathbb{P}$," but better would be for some specific forcing notions. Best would be to characterize the pairs $(T,\mathbb{P})$ for which we have such models. The results say (e.g.) that there are models $M_1,M_2$ which are not isomorphic (and even \emph{far} from being isomorphic, in a rigorous sense) which become isomorphic when we extend the universe by adding a new branch to the tree $({}^{\theta>}2,\lhd)$. We shall mention some specific choices of $\mathbb{P}$: mainly $({}^{\theta>}2,\lhd)$ with $\theta = \theta^{<\theta}$. That is, the reader just has to agree that starting with a universe $\mathbf{V}$ of set theory (i.e.~a model of $ZFC$) and a quasiorder $\mathbb{P}$, there are new directed $\mathbf{G} \subseteq \mathbb{P}$ meeting every dense subset of $D$ of $\mathbb{P}$, a universe $\mathbf{V}[\mathbf{G}]$ (so {it} satisfies $ZFC$) of which the original $\mathbf{V}$ is a transitive subclass. We may say that $\mathbf{V}[\mathbf{G}]$ (also denoted $\mathbf{V}^\mathbb{P}$) is the universe obtained by forcing with $\mathbb{P}$. This work does not require any serious knowledge of forcings, nor of stability theory. This is part of the classification and {Main Gap} program.