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Theories with models of prescribed cardinalities1

Published online by Cambridge University Press:  12 March 2014

Alan Mekler
Alan Mekler*
Affiliation:
Lakehead University, Thunder Bay, Ontario
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Extract

The Löwenheim–Skolem theorem states that if a theory has an infinite model it has models of all cardinalities greater than or equal to the cardinality of the language in which the theory is defined. A natural question is what happens if there is a model whose cardinality is less than that of the language.

If κ is an infinite cardinal less than the first measurable cardinal and κ < κω, the Rabin–Keiler theorem [1, p. 139] gives an example of a theory which has a model of cardinality κ in which every element is the interpretation of a constant and all other models have cardinality μ ≥ κω. Keisler has also shown that if a theory has a model of cardinality κ it has models of all cardinalities μ ≥ κω. We will show that within the bounds of the above theorems anything can happen.

The main result is as follows.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 2 , June 1977 , pp. 251 - 253
Copyright
Copyright © Association for Symbolic Logic 1977

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Footnotes

1

Research supported by National Research Council of Canada under grant numbers A8599, A5603 and A8190.

References

BIBLIOGRAPHY

[1]Bell, J. L. and Slomson, A. B., Models and ultraproducts, North-Holland, Amsterdam, 1969.Google Scholar
[2]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[3]Rabin, M. O., Arithmetical extensions with prescribed cardinality, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings. Series A 62 = Indagationes Malhematicae, vol. 21 (1959), pp. 439–446.Google Scholar