Logic Seminar

Shaun Allison University of Toronto
Automorphisms with Knight rank and bounds on cardinalities of potential Scott sentences

Friday, April 25, 2025 - 2:55pm
Malott 205

In recent work I developed the Knight model rank of a countable structure M, inspired by a construction by Julia Knight called "Knight's model". The Knight model rank identifies when the structure's automorphism group ${\rm Aut}(M)$ involves $S_\infty$ and thus has "maximal classification strength" in the sense of invariant descriptive set theory. We develop this rank further to show that given a bound on the Knight model rank of $M$, we can bound the cardinality of any potential Scott sentence for a model extending $M$. The talk will include a brief review of Scott sentences and their potential versions, as well as some historical context behind Knight's original construction that significantly motivated this work.