Counting partial types in simple theories

Olivier Lessmann

Colloquium Mathematicae (2000)

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

Abstract

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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.

How to cite

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Lessmann, 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

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  1. [Ca] E. Casanovas, The number of types in simple theories, Ann. Pure Appl. Logic 98 (1999), 69-86. Zbl0939.03039
  2. [GIL] R. Grossberg, J. Iovino, and O. Lessmann, A primer of simple theories, preprint. 
  3. [Ke] H. J. Keisler, Six classes of theories, J. Austral. Math. Soc. 21 (1976), 257-256. Zbl0342.02035
  4. [K] B. Kim, Forking in simple unstable theories, J. London Math. Soc. 57 (1998), 257-267. Zbl0922.03048
  5. [KP] B. Kim and A. Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), 149-164. Zbl0897.03036
  6. [Sh a] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, rev. ed., North-Holland, 1990. 
  7. [Sh] S. Shelah, Simple unstable theories, Ann. Math. Logic 19 (1998), 177-203. Zbl0489.03008

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