On ground model definability

TL;DR

This paper explores the conditions under which models of various set theories are definable in their forcing extensions, extending previous results and demonstrating both positive and negative cases of ground model definability.

Contribution

It generalizes ground model definability results to models of ZF+DC_delta and ZFC^- and shows limitations for models of ZFC^- like H_{kappa^+}.

Findings

Models of ZF+DC_delta are uniformly definable in their forcing extensions by posets with a gap at delta.

It is consistent that models of ZFC^- such as H_{kappa^+} are not definable in their Cohen forcing extensions.

There exists a ZFC universe where a large cardinal H_{kappa^+} is not definable in its forcing extension.

Abstract

Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are uniformly definable in their set-forcing extensions, using a ground model parameter. We investigate ground model definability for models of fragments of ${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_\delta$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_\delta$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $\delta$, using a ground model parameter. In particular, this means that models of ${\rm ZF}+{\rm DC}_\delta$ are uniformly definable in their forcing extensions by posets of size less than $\delta$. We also show that it is consistent for ground model definability to fail for models of ${\rm ZFC}^-$ of the form $H_{\kappa^+}$. Using forcing, we produce a ${\rm ZFC}$ universe in which there is a cardinal $\kappa>\!>\omega$ such that $H_{\kappa^+}$ is not definable in its Cohen forcing extension. As a corollary, we show that there is always a countable transitive model of ${\rm ZFC}^-$ violating ground model definability. These results turn out to have a bearing on ground model definability for models of ${\rm ZFC}$. It follows from our proof methods that the hereditary size of the parameter that Woodin used to define a ${\rm ZFC}$ model in its set-forcing extension is best possible.