November 19
Bartosz Wcisło,
University of Gdańsk
Saturation properties for propositionally sound satisfaction classes
Over the last years, a lot of progress has been achieved in understanding the arithmetical strength of axiomatic theories of compositional truth. It turned out that a theory $\mathsf{CT}^-$ of compositional truth for arithmetical sentences can become non-conservative over $\mathsf{PA}$ upon adding some seemingly benign principles.
One of the principles whose arithmetical strength is still unknown is the axiom of propositional soundness which says that for any arithmetical sentence $\phi$ which is a propositional tautology, $\phi$ is true in the sense of the truth predicate. It is an open problem whether this axiom together with $CT^-$ is conservative over $PA$.
In our talk, we will show that if $(M,T)$ is a model of $\mathsf{CT}^-$ satisfying the propositional soundness principle, then $(M,T)$ satisfies a certain amount of saturation: if $(\phi_i)_{i \lt c}$ is a sequence of sentences such that for any standard $i$, $\phi_i$ is true in the sense of the truth predicate, then there is a nonstandard $d$ such that for each $i \in [0,d]$, $\phi_i$ is true. This puts very strong limitations on any possible conservativeness proof. The result may be seen as a counterpart to the classical theorem of Lachlan which says that the arithmetical part of any model of $\mathsf{CT}^-$ is recursively saturated.