Kim-independence in positive logic

Jan Dobrowolski, Mark Kamsma

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An important dividing line in the class of unstable theories is being NSOP1, which is more general than being simple. In NSOP1 theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP1 theories to positive logic—a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP1 and various other properties. We prove that, in a thick positive NSOP1 theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in a first-order NSOP1 theory. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP1 by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable.

In first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in first-order theories. In particular, we only work with trees of finite height.
Original languageEnglish
Pages (from-to)55-113
Number of pages59
JournalModel Theory
Issue number1
Publication statusPublished - 24 Jun 2022


  • Kim-independence
  • Kim-dividing
  • positive logic
  • NSOP1 theory

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