**May 22**

**Ali Enayat**,
University of Gothenburg

**Recursively saturated models of set theory and their close relatives: Part II**

A model $M$ of set theory is said to be 'condensable' if there is an 'ordinal' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is both isomorphic to $M$, and an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M$. Clearly if $M$ is condensable, then $M$ is ill-founded. The work of Barwise and Schlipf in the mid 1970s showed that countable recursively saturated models of ZF are condensable.

In this two-part talk, we present a number of new results related to condensable models, including the following two theorems.

Theorem 1. Assuming that there is a well-founded model of ZFC plus 'there is an inaccessible cardinal', there is a condensable model $M$ of ZFC which has the property that every definable element of $M$ is in the well-founded part of $M$ (in particular, $M$ is $\omega$-standard, and therefore not recursively saturated).

Theorem 2. The following are equivalent for an ill-founded model $M$ of ZF of any cardinality:

(a) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension.

(b) There is a cofinal subset of 'ordinals' $\alpha$ of $M$ such that the rank initial segment of $M$ determined by $\alpha$ is an elementary submodel of $M$ for infinitary formulae appearing in the well-founded part of $M.$

Moreover, if $M$ is a countable ill-founded model of ZFC, then conditions (a) and (b) above are equivalent to:

(c) $M$ is expandable to Gödel-Bernays class theory plus $\Delta^1_1$-Comprehension + $\Sigma^1_1$-Choice.