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On theories categorical in their own power

Published online by Cambridge University Press:  12 March 2014

H. Jerome Keisler
H. Jerome Keisler*
Affiliation:
University of Wisconsin, Madison, Wisconsin
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Extract

A theory T is said to be categorical in power κ iff T has a model of power κ and any two models of power κ are isomorphic.

It was conjectured by Morley [4] that if T is a theory in a language with κ > ω symbols and T is categorical in power κ, then T has a model of power < κ. The aim of this paper is to prove the following theorem.

Theorem A. Let κ be a regular cardinal such that ω < κ < 2ω. Let T be a theory in a language with κ symbols such that T is categorical in power κ. Then:

(a) T has a model of power < κ.

(b) T is categorical in all powers μ ≥ κ.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 240 - 244
Copyright
Copyright © Association for Symbolic Logic 1971

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References

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