June 24
Bartosz Wcisło,
Polish Academy of Sciences
Tarski boundary III
Truth theories investigate the notion of truth using axiomatic methods. To a fixed base theory (typically Peano Arithmetic ${\rm PA}$) we add a unary predicate $T(x)$ with the intended interpretation '$x$ is a (code of a) true sentence.' Then we analyse how adding various possible sets of axioms for that predicate affects its behaviour.
One of the aspects which we are trying to understand is which truth-theoretic principles make the added truth predicate 'strong' in that the resulting theory is not conservative over the base theory. Ali Enayat proposed to call this demarcating line between conservative and non-conservative truth theories 'the Tarski boundary.'
Research on Tarski boundary revealed that natural truth theoretic principles extending compositional axioms tend to be either conservative over ${\rm PA}$ or exactly equivalent to the principle of global reflection over $ A$. It says that sentences provable in ${\rm PA}$ are true in the sense of the predicate $T$. This in turn is equivalent to $\Delta_0$ induction for the compositional truth predicate which turns out to be a surprisingly robust theory.
The equivalences between nonconservative truth theories are typically proved by relatively direct ad hoc arguments. However, certain patterns seem common to these proofs. The first one is construction of various arithmetical partial truth predicates which provably in a given theory have better properties than the original truth predicate. The second one is deriving induction for these truth predicates from internal induction, a principle which says that for any arithmetical formula, the set of those elements for which that formula is satisfied under the truth predicate satisfies the usual induction axioms.
As an example of this phenomenon, we will present two proofs. First, we will show that global reflection principle is equivalent to local induction. Global reflection expresses that any sentence provable in ${\rm PA}$ is true. Local induction says that any predicate obtained by restricting truth predicate to sentences of a fixed syntactic complexity $c$ satisfies full induction. This is an observation due to Mateusz Łełyk and the author of this presentation.
The second example is a result by Ali Enayat who showed that ${\rm CT}_0$, a theory compositional truth with $\Delta_0$ induction, is arithmetically equivalent to the theory of compositional truth together with internal induction and disjunctive correctness.
This talk is intended as a continuation of 'Tarski boundary II' presentation at the same seminar. However, we will try to avoid excessive assumptions on familiarity with the previous part.