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.
Bibliographical noteFunding Information:
The first author was supported by the Marie Curie Initial Training Network in Mathematical Logic - MALOA - From MAthematical LOgic to Applications, PITN-GA-2009-238381. The second author was partially supported by the Independent Research Start-up Grant in the Zukunftskolleg (University of Konstanz), the SFB grant 878 (University of Münster) and the Israel Science Foundation (grant No. 1533/14). The third author has received funding from the European Research Council, ERC Grant Agreement no. 338821. He would like to thank the Israel Science Foundation for partial support of this research (Grants 710/07 and 1053/11). No. 1007 on the third author's list of publications.
© European Mathematical Society 2016.
- Cardinal arithmetic
- Dedekind cuts