ISSN:
1432-0665
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Mathematik
Notizen:
Abstract Models which omit a fixed set of types are considered, and notions analogous to those of saturation and compactness are formulated in this context. Conditions are found under which such saturated models exist and are unique; and the preservation of compactness under direct products and homomorphisms is proven. Atomic and positive compactness are shown to be equivalent in this context. In [5] Morley and Vaught give conditions on a class of structuresK which ensure thatK has homogeneous universal models of certain powers. The work is based on the nature of the structures concerned and not on that of the language for which they are assumed to be models. Keisler [2] and [3] considered the notion of a saturated model and showed that for certain cardinalsκ, a complete theoryT in elementary logic has a saturated model of powerκ, which is homogeneous-universal in the sense of Morley and Vaught ifT is neat. Keisler's definition of saturated model is intimately connected with the language involved, and the results are dependent on the semantics of that language. In this paper we consider such problems as the existence of “saturated” models, and under what conditions they are homogeneous-universal, assuming that a specified set of types is omitted by all the models under consideration. In § 1 we formulate the definition of aλ-(ℒ, J, K)-saturated model forλ a cardinal,ℒ a set of types,J andK sets of formulae, and prove some basic results. In § 2 the existence ofλ +-(ℒ, J, K.)-saturated models of power 2 λ is proven and in § 3 a condition is given for the uniqueness of such a model. We examine (ℒ, J, K)-special models in § 4. This work is based on that of Bell and Slomson [1] and Wilmers [10]. In § 5λ-(ℒ, J)-K compact models are defined and we examine closure under direct products and homomorphisms. Lastly, in § 6λ-(ℒ, J)-atomic compactness andλ-(ℒ, J)-positive compactness are shown to be equivalent.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/BF02007252
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