November 7
Emma Palmer,
University of Oxford
Tarski's Revenge
Tarski’s nondefinability theorem tells us that we cannot internally define satisfaction in a model via a single first-order formula. But how close can we get to that situation without running into inconsistency? We explore a variety of axioms about satisfaction classes, which, surprisingly, all turn out to be equiconsistent and have only mild consistency strength. For example, from a model of ZFC with an inaccessible cardinal, we can obtain a model of GBC with a definable satisfaction class for an inner model. Indeed, this inner model can even be HOD, or the mantle. Finally, we consider the statement that any set is contained in an inner model with a definable satisfaction class — an axiom we call 'Tarski's Revenge'.