# 2015 seminar talk: Satisfaction in outer models

Talk held by Radek Honzik (KGRC) at the KGRC seminar on 2015-03-26.

### Abstract

We will study a generalized notion of satisfaction in which the collection of test structures is restricted to outer models of a given transitive set model $M$ of ZFC. We will show that it is consistent from an inaccessible cardinal that there is $M$ which can define in a lightface way satisfaction in its outer models (we say that $M$ defines its outer model theory). The proof uses Barwise's results on infinitary logic $L_{\infty,\omega}$ and a non-monotonic Easton-type iteration which manipulates the continuum function on regular cardinals in $M$ and which is longer (in terms of ordinal type) than the number of ordinals in $M$. The result complements an unpublished result of Mac Stanley who showed that if $M$ contains many Ramsey cardinal then it defines its outer model theory.

The work is joint with Sy Friedman.