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.

Slides

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe. The main question is: can there be an embedding $j:Vto L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $Vneq L$? The notion of embedding here is merely that $xin y$ if and only if $j(x)in j(y)$, and such a map need not be elementary nor even $Delta_0$-elementary. It is not difficult to see that there can generally be no $Delta_0$-elementary embedding $j:Vto L$, when $Vneq L$.  Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, which shows that every countable model $M$ does admit an embedding $j:Mto L^M$ into its constructible universe. More generally, any two countable models of set theory are comparable; one of them embeds into the other. Indeed, one model $langle M,in^Mrangle$ embeds into another $langle N,in^Nrangle$ just in case the ordinals of the first $text{Ord}^M$ order-embed into the ordinals of the second $text{Ord}^N$.  In these theorems, the embeddings $j:Mto L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. Currently, the question remains open, but we have some partial progress, settling it in a number of cases.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut.  See more information at the links below:

Blog post for this talk |  Related MathOverflow question | Article

The speaker will give the second part of her talk, continued from the previous week. An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.

An $\omega_1$-like model of set theory is an uncountable model, all of whose initial segments are countable. The speaker will present two $\omega_1$-like models of set theory, constructed using $\Diamond$, which are incomparable with respect to embeddability: neither is isomorphic to a submodel of the other. Under a suitable large cardinal assumption, there are such models that are well-founded.