# Ehrenfeucht principles in set theory

## Victoria Gitman

### The City University of New York

A powerful tool in the field of models of Peano Arithmetic (${\rm PA}$) is Ehrenfeucht’s lemma, due to Ehrenfeucht, which states that if $a\neq b$ are elements of a model $M$ of ${\rm PA}$ such that $b$ is definable from $a$ in $M$, then $a$ and $b$ must have different types in $M$. Considering the active and fruitful interchange of ideas that has historically existed between the fields of models of ${\rm PA}$ and of models of set theory, it is natural to wonder whether Ehrenfeucht’s lemma holds for models of ${\rm ZFC}$ (or even ${\rm ZF}$). Ehrenfeucht’s original argument generalizes to show that Ehrenfeucht’s lemma holds in every model of ${\rm ZF}+V={\rm HOD}$. I will show that Ehrenfeucht’s lemma fails in a very strong sense in any Cohen forcing extension. I will then introduce the more general *Ehrenfeucht principles*, which are parametric generalizations of Ehrenfeucht’s lemma, and argue that they unify several interesting and already deeply studied model-theoretic/set-theoretic principles for models of set theory. Finally, I will discuss some open questions surrounding this topic involving a connection between Ehrenfeucht principles and global choice principles. This is joint work with Gunter Fuchs and Joel David Hamkins.

Victoria Gitman received her Ph.D. in 2007 from the CUNY Graduate Center, as a student of Joel Hamkins, and is presently a visiting scholar at the CUNY Graduate Center. Her research is in Mathematical Logic, in particular in the areas of Set Theory and Models of Peano Arithmetic.