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Compactly expandable models and stability

Published online by Cambridge University Press:  12 March 2014

Enrique Casanovas*
Affiliation:
Departamento de Lógica, Historia y Filosofía de la Ciencia, Universidad de Barcelona, 08028 Barcelona, Spain, E-mail: [email protected]

Extract

In analogy to ω-logic, one defines M-logic for an arbitrary structure M (see [5],[6]). In M-logic only those structures are considered in which a special part, determined by a fixed unary predicate, is isomorphic to M. Let L be the similarity type of M and T its complete theory. We say that M-logic is κ-compact if it satisfies the compactness theorem for sets of < κ sentences. In this paper we introduce the related notion of compactness for expandability: a model M is κ-compactly expandable if for every extension T′T of cardinality < κ, if every finite subset of T′ can be satisfied in an expansion of M, then T′ can also be satisfied in an expansion of M. Moreover, M is compactly expandable if it is ∥M+-compactly expandable. It turns out that M-logic is κ-compact iff M is κ-compactly expandable.

Whereas for first-order logic consistency and finite satisfiability are the same, consistency with T and finite satisfiability in M are, in general, no longer the same thing. We call the model Mκ-expandable if every consistent extension T′ ⊇ T of cardinality < κ can be satisfied in an expansion of M. We say that M is expandable if it is ∥M+-expandable. Here we study the relationship between saturation, expandability and compactness for expandability. There is a close parallelism between our results about compactly expandable models and some theorems of S. Shelah about expandable models, which are in fact expressed in terms of categoricity of PC-classes (see [7, Th. VI.5.3, VI.5.4 and VI.5.5]). Our results could be obtained directly from these theorems of Shelah if expandability and compactness for expandability were the same notion.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

[1] Baldwin, John T. and Kueker, David W., Ramsey quantifiers and the finite cover property, Pacific Journal of Mathematics, vol. 90 (1980), pp. 1119.CrossRefGoogle Scholar
[2] Keisler, H. JeromeUltraproducts which are not saturated, this Journal, vol. 32 (1967), pp. 2346.Google Scholar
[3] Kojman, Menachem and Shelah, SaharonNonexistence of universal orders in many cardinals, this Journal, vol. 57 (1992), pp. 875891.Google Scholar
[4] Kojman, Menachem and Shelah, Saharon, The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic, vol. 58 (1992), pp. 5772.CrossRefGoogle Scholar
[5] Morley, MichaelThe Lowenheim-Skolem theorem for models with standard part, Symposia Mathematica, vol. 5 (1970), pp. 4352.Google Scholar
[6] Morley, MichaelCountable models with standard part, Logic, methodology and philosophy of science iv (Suppes, , Moisil, , and Joja, , editors), North Holland P.C., 1973.Google Scholar
[7] Shelah, Saharon, Classification theory, North Holland P.C., Amsterdam, 1978.Google Scholar
[8] Shelah, SaharonIndependence results, this Journal, vol. 45 (1980), pp. 563573.Google Scholar