Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T10:14:03.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Diverse classes

Published online by Cambridge University Press:  12 March 2014

John T. Baldwin
John T. Baldwin*
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60680
Get access Rights & Permissions [Opens in a new window]

Abstract

Let I(μ, Κ) denote the number of nonisomorphic models of power μ and IE(μ, Κ) the number of nonmutually embeddable models. We define in this paper the notion of a diverse class and use it to prove a number of results. The major result is Theorem B: For any diverse class Κ and μ greater than the cardinality of the language of Κ,

From it we deduce both an old result of Shelah, Theorem C: If T is countable and λ0 > ℵ0 then for every μ > ℵ0, IE(μ, T) ≥ min(2μ, ⊐2), and an extension of that result to uncountable languages, Theorem D: If ∣T∣ < 2ωλ0 > ∣T∣, and ∣D(T)∣ = ∣T∣ then for μ > ∣T∣,

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 3 , September 1989 , pp. 875 - 893
Copyright
Copyright © Association for Symbolic Logic 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Baldwin, J. T. [1988], Fundamentals of stability theory, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Chang, C. C. and Keisler, H. J. [1973], Model theory, North-Holland, Amsterdam.Google Scholar
Ebbinghaus, H.-D. [1985], Extended logics: the general framework, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, pp. 25–76.Google Scholar
Morley, M. [1968], Partitions and models, Proceedings of the summer school in logic (Loeb, M. H., editor), Springer-Verlag, Berlin, pp. 109–158.Google Scholar
Shelah, S. [1978], Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam.Google Scholar
Shelah, S. [1989], The number of pairwise non-elementarily-embeddable-models, this Journal, vol. 54 (to appear).Google Scholar