May 27
Bartosz Wcisło,
Polish Academy of Sciences
Tarski boundary II
Truth theories investigate the notion of truth with 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 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 ${\rm PA}$. 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.
In our talk, we will try to sketch proofs representative of research on Tarski boundary. We will present the proof by Enayat and Visser showing that the compositional truth predicate is conservative over ${\rm PA}$. We will also try to discuss how this proof forms a robust basis for further conservativeness results.
On the non-conservative side of Tarski boundary, the picture seems less organised, since more arguments are based on ad hoc constructions. However, we will try to show some themes which occur rather repeatedly in these proofs: iterated truth predicates and the interplay between properties of good truth-theoretic behaviour and induction. To this end, we will present the argument that disjunctive correctness together with the internal induction principle for a compositional truth predicate yields the same consequences as $\Delta_0$-induction for the compositional truth predicate (as proved by Ali Enayat) and that it shares arithmetical consequences with global reflection. The presented results are currently known to be suboptimal.
This talk is intended as a continuation of 'Tarski boundary' presentation. However, we will try to avoid excessive assumptions on familiarity with the previous part.