The Scandinavian Logic Society

NOL Seminar with Albert Visser

The Nordic Online Logic Seminar is organised monthly over Zoom, with expository talks on topics of interest for the broader logic community. The seminar is open for professional or aspiring logicians and logic aficionados worldwide. If you wish to receive the Zoom ID and password for it, as well as further announcements, please subscribe here:

Date: Monday, 25 March 2024 at 16:00 CET (UTC+1) on Zoom
Speaker: Albert Visser, Professor of Logic at Utrecht University
Title: Restricted Sequential Theories

Sequential theories form a fundamental class of theories in logic. They have full coding possibilities and allow us to build partial satisfaction predicates for formulas of bounded depth-of-quantifier-alernations. In many respects, they are the proper domain of Gödelian metamathematics. We explain the notion of sequential theory.

A theory is restricted if it can be axiomatised by axioms of bounded depth-of-quantifier-alernations. All finitely axiomatised theories are restriced, but, for example, also Primitive Recursive Arithmetic. We explain the small-is-very-small principle for restricted sequential theories which says that, whenever the given theory shows that a definable property has a small witness, i.e., a witness in a sufficiently small definable cut, then it shows that the property has a very small witness, i.e., a witness below a given standard number.

We sketch two proofs that restricted theories are incomplete (however complex the axiom set). One uses the small-is-very-small principle and the other a direct Rosser argument. (The second argument was developed in collaboration with Ali Enayat.)