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## Colloquium Mathematicum

2000 | 83 | 2 | 201-208
Tytuł artykułu

### Counting partial types in simple theories

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EN
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EN
We continue the work of Shelah and Casanovas on the cardinality of families of pairwise inconsistent types in simple theories. We prove that, in a simple theory, there are at most $λ^{<κ(T)} + 2^{μ +|T|}$ pairwise inconsistent types of size μ over a set of size λ. This bound improves the previous bounds and clarifies the role of κ(T). We also compute exactly the maximal cardinality of such families for countable, simple theories. The main tool is the fact that, in simple theories, the collection of nonforking extensions of fixed size of a given complete type (ordered by reverse inclusion) has a chain condition. We show also that for a notion of dependence, this fact is equivalent to Kim-Pillay's type amalgamation theorem; a theory is simple if and only if it admits a notion of dependence with this chain condition, and furthermore that notion of dependence is forking.
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Tom
Numer
Strony
201-208
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-06
poprawiono
1999-08-07
Twórcy
autor
• Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607, U.S.A.
Bibliografia
• [Ca] E. Casanovas, The number of types in simple theories, Ann. Pure Appl. Logic 98 (1999), 69-86.
• [GIL] R. Grossberg, J. Iovino, and O. Lessmann, A primer of simple theories, preprint.
• [Ke] H. J. Keisler, Six classes of theories, J. Austral. Math. Soc. 21 (1976), 257-256.
• [K] B. Kim, Forking in simple unstable theories, J. London Math. Soc. 57 (1998), 257-267.
• [KP] B. Kim and A. Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), 149-164.
• [Sh a] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, rev. ed., North-Holland, 1990.
• [Sh] S. Shelah, Simple unstable theories, Ann. Math. Logic 19 (1998), 177-203.
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