Some model theory for axiomatic theories of truth (Roman Kossak)

The Logic and Metaphysics Workshop will meet on February 5th from 4:15-6:15 in-person at the Graduate Center (Room 7395) for a talk by Roman Kossak (CUNY).

Title: Some model theory for axiomatic theories of truth

Abstract: Tarski’s arithmetic is the complete theory of (N,+,x,Tr), where (N,+,x) is the standard model of arithmetic and Tr is the set of Gödel numbers of all true arithmetic sentences. An axiomatic theory of truth is an axiomatic subtheory of Tarski’s arithmetic. If (M,+,x,T) is a model of an axiomatic theory of truth, then we call T a truth class. In 1981, Kotlarski, Krajewski, and Lachlan proved that every completion of Peano’s arithmetic has a model that is expandable to a model  with a truth class T that satisfies all biconditionals in Tarski’s definition of truth formalized in PA. If T is such a truth class, it assigns truth values to all sentences in the sense of M, standard and nonstandard. The proof showed  that such truth classes can be quite pathological. For example, they may declare true some infinite disjunctions of the single sentence (0=1). In 2018, Enayat and Visser gave  a much simplified model-theoretic proof, which opened the door for further investigations of nonstandard truths, and many interesting new results by many authors appeared. I will survey some of them, concentrating on their model-theoretic content.

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