February 14
Bartosz Wcisło,
University of Warsaw
Tarski boundary
Our talk concerns axiomatic theories of truth predicates. They are theories obtained by adding to Peano Arithmetic (${\rm PA}$) a fresh predicate $T(x)$ with the intended reading '$x$ is (a code of) a true sentence in the language of arithmetic' together with some axioms governing newly added predicate.
The canonical example of such a theory is ${\rm CT}^-$ (Compositional Truth). Its axioms state that the truth predicate is compositional. For instance, a conjunction is true iff both conjuncts are. If we add to ${\rm CT}^-$ full induction in the extended language, we call the resulting theory ${\rm CT}$.
It is easy to check that ${\rm CT}$ is not conservative over ${\rm PA}$, i.e., it proves new arithmetical sentences. On the other hand, by a nontrivial theorem of Kotlarski, Krajewski, and Lachlan, ${\rm CT}^-$ extends ${\rm PA}$ conservatively.
In our talk, we will discuss results on the strength of theories between ${\rm CT}^-$ and ${\rm CT}$. It turns out that the natural axioms concerning purely truth theoretic properties of the newly added predicate (as opposed to axiom schemes which are consequences of induction in more general context) are typically either conservative or exactly equal to ${\rm CT}_0$, the theory of compositional truth with $\Delta_0$-induction. Thus ${\rm CT}_0$ turns out to be a surprisingly robust theory and, arguably, the minimal 'natural' non-conservative theory of truth.