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Finite satisfiability and ℵ0-categorical structures with trivial dependence

Published online by Cambridge University Press:  12 March 2014

Marko Djordjević
Marko Djordjević*
Affiliation:
Department of Mathematics, Uppsala University, Box 480 75106 Uppsala, Sweden.E-mail:marko@math.uu.se
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Extract

The main subject of the article is the finite submodel property for ℵ0-categorical structures, in particular under the additional assumptions that the structure is simple, 1-based and has trivial dependence. Here, a structure has the finite submodel property if every sentence which is true in the structure is true in a finite substructure of it. It will be useful to consider a couple of other finiteness properties, related to the finite submodel property, which are variants of the usual concept of saturation.

For the rest of the introduction we will assume that M is an ℵ0-categorical (infinite) structure with a countable language. We also assume that there is an upper bound to the arity of the function symbols in M:s language and that, for every 0 < n < ℵ0 and R ⊆ Mn which is definable in M without parameters, there exists a relation symbol, in the language of M, which is interpreted as R; these assumptions are not necessary for most results to be presented, but it simplifies the statement of a result which I mention in this introduction.

First we will consider ‘canonically embedded’ substructures of Meq. Here, a structure N is canonically embedded in Meq if N's universe is a subset of Meq which is definable without parameters and, for every 0 < n < ℵ0 and R ⊆ Nn which is ε-definable in Meq there is a relation symbol in the language of N which is interpreted as R; we also assume that the language of N has no other relation (or function or constant) symbols.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 810 - 830
Copyright
Copyright © Association for Symbolic Logic 2006

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References

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