Dividing lines between positive theories

Anna Dmitrieva, Francesco Gallinaro, Mark Kamsma

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

We generalise the properties OP, IP, k-TP, TP1, k-TP2, SOP1, SOP2 and SOPto positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having TP and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has OP iff it has IP or SOP1 and that T has TP iff it has SOP1 or TP2, analogous to the well-known results in full first-order logic where SOP1 is replaced by SOP in the former and by TP1 in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.
Original languageEnglish
JournalJournal of Symbolic Logic
Early online date6 Dec 2023
DOIs
Publication statusE-pub ahead of print - 6 Dec 2023

Keywords

  • classification theory
  • dividing lines
  • independence property
  • order property
  • positive logic
  • simplicity
  • stability
  • strict order property
  • tree property

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