March 21
Bartosz Wcisło,
University of Gdańsk
Satisfaction classes with the full collection scheme: Part II
Satisfaction classes are subsets of models of Peano arithmetic which satisfy Tarski's compositional clauses. Alternatively, we can view satisfaction or truth classes as the extension of a fresh predicate T(x) (the theory in which compositional clauses are viewed as axioms is called CT^-).
It is easy to see that CT^- extended with a full induction scheme is not conservative over PA, since it can prove, for instance, the uniform reflection over arithmetic. By a nontrivial argument of Kotlarski, Krajewski, and Lachlan, the sole compositional axioms of CT^- in fact form a conservative extension of PA. Moreover, in order to obtain non-conservativity it is enough to add induction axioms for the Delta_0 formulae containing the truth predicate.
Answering a question of Kaye, we will show that the theory of compositional truth, CT^- with the full collection scheme is a conservative extension of Peano Arithmetic. Following the initial suggestion of Kaye, we will in fact show that any countable recursively saturated model M of PA has an elementary omega_1-like end extension M' such that M' carries a full satisfaction class.