- journal article metadata
European Mathematical Society Publishing House
2016-11-17 23:45:01
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
18
2016
12
On non-forking spectra
Artem
Chernikov
University of California at Los Angeles, LOS ANGELES, UNITED STATES
Itay
Kaplan
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Saharon
Shelah
The Hebrew University of Jerusalem, JERUSALEM, ISRAEL
Forking, dividing, NIP, NTP2, circularization, Dedekind cuts, cardinal arithmetic
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 $\kappa$ and $\lambda$ giving the supremum of the possible number of types over a model of size $\lambda$ that do not fork over a sub-model of size $\kappa$. 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 $\kappa$ < (ded$_\kappa) ^{\omega}$.
Mathematical logic and foundations
2821
2848
10.4171/JEMS/654
http://www.ems-ph.org/doi/10.4171/JEMS/654