On ground model definability

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 ${\rm ZF}+{\rm DC}_δ$ and of ${\rm ZFC}^-$, and we obtain both positive and negative results. Generalizing the results of Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_δ$ are uniformly definable in their set-forcing extensions by posets admitting a gap at $δ$, using a ground model parameter. In particular, this means that models of ${\rm ZF}+{\rm DC}_δ$ are uniformly definable in their forcing extensions by posets of size less than $δ$. We also show that it is consistent for ground model definability to fail for models of ${\rm ZFC}^-$ of the form $H_{κ^+}$. Using forcing, we produce a ${\rm ZFC}$ universe in which there is a cardinal $κ>\!>ω$ such that $H_{κ^+}$ 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.