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On the existence of atomic models

Published online by Cambridge University Press:  12 March 2014

M. C. Laskowski
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, E-mail: mcl@math.umd.edu
S. Shelah
Affiliation:
Department of Mathematics, The Hebrew University, Jerusalem, Israel Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Abstract

We give an example of a countable theory T such that for every cardinal λ ≥ ℵ2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size λ where the principal types are dense, yet T(A) has no atomic model.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1993

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References

[1]Erdös, P., Hajnal, A., Mate, A., and Rado, P., Combinatorial set theory, North-Holland, Amsterdam, 1984.Google Scholar
[2]Knight, J., Prime and atomic models, this Journal, vol. 43 (1978), pp. 385393.Google Scholar
[3]Kueker, D. W., Uniform theorems in infinitary logic, Logic Colloquium '77 (Macintyre, A., Pacholski, L., and Paris, J., editors), North-Holland, Amsterdam, 1978.Google Scholar
[4]Kueker, D. W. and Laskowski, M. C., On generic structures, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 175183.CrossRefGoogle Scholar
[5]Shelah, S., Classification theory, North-Holland, Amsterdam, 1978.Google Scholar
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