Logic Seminar - Mark Kamsma

Wed, 16/12/202011:15-13:00

Kim-independence in positive logic

Abstract: This is joint work (in progress) with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where we only allow the logical connectives for conjunction, disjunction, falsehood, truth and the existential quantifier. For example, if we add hyperimaginaries as parameters to our monster model then we leave the framework of first-order logic, but we remain in the framework of positive logic. Another interesting example is the positive theory of existentially closed exponential fields (ECEF), introduced by Haykazyan and Kirby.
    There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence. We have generalised the work on Kim-independence to the setting of positive logic. In this talk I will explain the challenges that positive logic presents us and how we solve them. Our result can then be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-dividing.
    Haykazyan and Kirby proved that ECEF has a nice enough independence relation and showed that it is thus NSOP1. So our theorem applies and confirms that this independence relation is given by Kim-dividing. The operation of adding hyperimaginaries as parameters preserves the NSOP1 property. So in particular, if we start with an NSOP1 first-order theory and we add hyperimaginaries then our theorem still applies.