Unary Interpretability Logic

Bi-Unary Interpretability Logic

Interpretability suprema in Peano Arithmetic

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic ILM of Peano Arithmetic (PA). It is wellknown that any theories extending PA have a supremum in the interpretability ordering. While provable in PA, this fact is not reflected in the theorems of the modal system ILM, due to limited expressive power. Our goal is to enrich the lan...

Interpretability Logic

This is an overview a study of interpretability logic in Zagreb for the last twenty years: a brief history and some planes for further research. The idea of treating a provability predicate as a modal operator goes back to Gödel. The same idea was taken up later by Kripke and Montague, but only in the mid–seventies was the correct choice of axioms, based on Löb’s theorem, seriously considered b...

A note on the interpretability logic of finitely axiomatized theories

Ill [6] Albert Visser shows that ILP completely axiomatizes all schemata about provabihty and relative interpretability that are prov-able in finitely axiomatized theories. In this paper we introduce a system called ILP ~ that completely axiomatizes the arithmetically valid principles of provability in and interpretabihty over such theories. To prove the arith-metical completeness of ILP ~ we u...

Fixed Point vs. First-Order Logic on Finite Ordered Structures with Unary Relations

We prove that rst order logic is strictly weaker than xed point logic over every in nite classes of nite ordered structures with unary relations: Over these classes there is always an inductive unary relation which cannot be de ned by a rst-order formula, even when every inductive sentence (i.e., closed formula) can be expressed in rst-order over this particular class. Our proof rst establishes...