Mateusz Łełyk, University of Warsaw
Model theoretic characterizations of truth: Part I
This is joint work (still in progress) with Bartosz Wcisło (who will give the second part of the talk). By an axiomatic theory of truth (for the language of arithmetic, $L$) we mean a theory in L enriched with a fresh unary predicate $T(x)$ which (extends the elementary arithmetic EA and) proves all sentences of the form ($\phi$ being a sentence in L) $T(\phi)\equiv \phi.$
The collection of all sentence of the above form is normally called $TB^-$. It is well known that axiomatic theories of truth have a number of interesting model-theoretic consequences. For example, already relatively weak theories of truth impose recursive saturation, in the sense that the L-reduct of any model of such theory is recursively saturated. To give another example, already $TB^-$ imposes elementary equivalence of models, in the sense that whenever $(M,T)\models TB^-$, $(M',T')\models TB^-$, and $(M,T)\subset (M', T)$ (the first model is a submodel of the second one), then actually $M$ and $M'$ are elementarily equivalent. During (both parts) of the talk we investigate which of these properties actually characterize the respective truth theory up to definability. In particular, in the first part of the talk, we prove the following results (we restrict ourselves to theories in a finite language and extending EA):
- Every theory which imposes elementary equivalence defines $TB^-$.
- Every theory which imposes full elementarity defines $UTB^-$.
Additionally, we take a look at the definability relations between axiomatic truth theories and axiomatic theories of definability or skolem functions.