# Blog Archives

# Topic Archive: Ehrenfeucht's lemma

# Ehrenfeucht principles in set theory

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.