Ali Enayat: Satisfaction classes: conservativity, interpretability, and speed-up

This talk reports on joint work with Albert Visser (Utrecht). A full satisfaction class on a model M of PA (Peano arithmetic) is a subset S of M that decides the 'truth' of all arithmetical formulae with parameters in M (including those of nonstandard length, if M is nonstandard), while obeying the usual recursive Tarski conditions for a satisfaction predicate. I will present a robust technique for building a wide variety of full satisfaction classes using model-theoretic ideas. This model-theoretic construction lends itself to certain arithmetizations, which in turn can be employed to show that the conservativity of PA + "S is a full satisfaction class" over PA can be verified in Primitive Recursive Arithmetic. I will also comment on the ramifications of the aforementioned arithmetization on interpretability, and speed-up of PA + "S is a full satisfaction class" over PA.