Mateusz Łełyk, University of Warsaw
Pathologies in Satisfaction Classes
This is the second part of the talk given by Athar Abdul-Quader (Pathologically definable subsets of models of CT-), however we will make sure to make it self-contained.
The talk is centered around the following problem: when a subset of a countable and recursively saturated model M can be characterized as the set of the lengths of disjunctions on which a satisfaction class behaves correctly? More precisely: let DC(x) denote a sentence in a language of PA with a fresh binary predicate S which says 'For every disjunction d with at most x disjuncts and every assignment a, S(d,a) iff there is a disjunct d' in d such that S(d',a).' We say that X is a DC-set in (M,S) iff X is precisely the set of those numbers a such that (M,S) satisfies DC(a). We ask: given a countable and recursively saturated model M for which subsets X of M we can find a satisfaction class S such that X is a DC-set in (M,S)?
In the talk we study this problem for idempotent disjunctions, that is: disjunctions which repeat the same sentence. Let IDC(x) be DC(x) restricted to such 'idempotent' disjunctions of length x. The following is one of our core results:
Theorem: For an arbitrary countable and recursively saturated model M of PA the following conditions are equivalent:
(a) M is arithmetically saturated
(b) For every cut I in M there is a satisfaction class S such that I is an IDC-set in (M,S).
We study how this result generalizes to other propositional constructions in the place of disjunctions. The talk is based on a joint work with Athar Abdul-Quader presented in this paper from arxiv: arXiv:2303.18069v1 [math.LO] 31 Mar 2023.