# Counting partial types in simple theories

Colloquium Mathematicae (2000)

- Volume: 83, Issue: 2, page 201-208
- ISSN: 0010-1354

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topLessmann, Olivier. "Counting partial types in simple theories." Colloquium Mathematicae 83.2 (2000): 201-208. <http://eudml.org/doc/210781>.

@article{Lessmann2000,

abstract = {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.},

author = {Lessmann, Olivier},

journal = {Colloquium Mathematicae},

keywords = {simple theory; partial types; chain condition},

language = {eng},

number = {2},

pages = {201-208},

title = {Counting partial types in simple theories},

url = {http://eudml.org/doc/210781},

volume = {83},

year = {2000},

}

TY - JOUR

AU - Lessmann, Olivier

TI - Counting partial types in simple theories

JO - Colloquium Mathematicae

PY - 2000

VL - 83

IS - 2

SP - 201

EP - 208

AB - 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.

LA - eng

KW - simple theory; partial types; chain condition

UR - http://eudml.org/doc/210781

ER -

## References

top- [Ca] E. Casanovas, The number of types in simple theories, Ann. Pure Appl. Logic 98 (1999), 69-86. Zbl0939.03039
- [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. Zbl0342.02035
- [K] B. Kim, Forking in simple unstable theories, J. London Math. Soc. 57 (1998), 257-267. Zbl0922.03048
- [KP] B. Kim and A. Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), 149-164. Zbl0897.03036
- [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. Zbl0489.03008

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