Curious satisfaction classes

We present two new constructions of satisfaction/truth classes over models of PA (Peano Arithmetic) that provide a foil to the fact that the existence of a disjunctively correct full truth class over a model M of PA implies that Con(PA) holds in M.Comment: 12 page

Truth, collection and deflationism in models of peano arithmetic

This thesis focuses on adding collection axioms to satisfaction classes and exploring the suitability of a formal deflationary truth predicate. Chapter 2 proves that every nonstandard, recursively saturated model of PA has a satisfaction class in which all collection axioms are true. Chapter 3 explores collection axioms for the language with the satisfaction predicate, ℒS, and proves that these entail the theory of chapter 2. This chapter then demonstrates a method of closing a model with a satisfaction class to produce a new model with an induced satisfaction class, which it is conjectured will not satisfy all Ʃ1 collection axioms in ℒS. In chapter 4 we conjecture that a new formulation of Visser and Enayat's construction of extensions of models with a satisfaction classes [5] will provide elementary extensions. Using this conjecture, we demonstrate new Tarski axioms provide satisfaction classes with Ʃ1 collection axioms and that these axioms can be built into the theory by reducing the language to one where formulas are stratified. Finally, in chapter 5 we argue for a new definition of a deflationary truth predicate and show that this entails there are no formalisations of a deflationary truth predicate for the full nonstandard language of arithmetic