Full satisfaction classes, definability, and automorphisms

TL;DR

This paper constructs full satisfaction classes in countable recursively saturated models of Peano Arithmetic, showing they can make all elements definable and analyzing their impact on automorphisms and definability.

Contribution

It proves the existence of full satisfaction classes that render all elements definable and explores their effects on automorphisms, extending prior partial results to full classes.

Findings

Existence of satisfaction classes making all elements definable

Automorphism groups are altered by satisfaction classes

Different elements can share types but differ in satisfaction class membership

Abstract

We show that for every countable recursively saturated model $M$ of Peano Arithmetic and every subset $A \subseteq M$, there exists a full satisfaction class $S_A \subset M^2$ such that $A$ is definable in $(M,S_A)$ without parametres. It follows that in every such model, there exists a full satisfaction class which makes every element definable and thus the expanded model is minimal and rigid. On the other hand, we show that for every full satisfaction class $S$ there are two elements which have the same arithmetical type, but exactly one of them is in $S$. In particular, the automorphism group of a model expanded with a satisfaction class is never equal to the automorphism group of the original model. The analogue of many of the results proved here for full satisfaction classes were obtained by Roman Kossak for partial inductive satisfaction classes. However, most of the proofs relied heavily on the induction scheme in a crucial way, so recapturing the results in the setting of full satisfaction classes requires quite different arguments.