On non-forking spectra

Artem Chernikov, Itay Kaplan, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum-a function of two cardinals κ and λ giving the supremum of the possible number of types over a model of size λ that do not fork over a submodel of size κ. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded κ < (ded κ)ω.

Original languageAmerican English
Pages (from-to)2821-2848
Number of pages28
JournalJournal of the European Mathematical Society
Volume18
Issue number12
DOIs
StatePublished - 2016

Bibliographical note

Publisher Copyright:
© European Mathematical Society 2016.

Keywords

  • Cardinal arithmetic
  • Circularization
  • Dedekind cuts
  • Dividing
  • Forking
  • NIP
  • NTP2

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