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The number of countable models

Published online by Cambridge University Press:  12 March 2014

Michael Morley
Michael Morley*
Affiliation:
Cornell University
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Extract

A theory formulated in a countable predicate calculus can have at most nonisomorphic countable models. It has been conjected (e.g., in [4]) that if it has an uncountable number of such models then it has exactly such. Of course, this would follow immediately if one assumed the continuum hypothesis.

In this paper we show that if a theory has more than ℵ1 (i.e., at least ℵ2) isomorphism types of countable models then it has exactly . Our results generalize immediately to theories in Lω1ω and even to pseudo-axiomatic classes in Lω1ω . In this last case, a result of H. Friedman shows that it is the best possible result.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 1 , March 1970 , pp. 14 - 18
Copyright
Copyright © Association for Symbolic Logic 1970

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References

[1] Friedman, H., On the ordinals in models of set theory, preprint.Google Scholar
[2] Kuratowski, K., Topology, Academic Press, New York, 1966.Google Scholar
[3] Scott, D., Logic with denumerably long formulas and finite strings of quantifiers in Theory of models, North-Holland, Amsterdam, 1965, pp. 329–341.Google Scholar
[4] Vaught, R. L., Denumerable models of complete theories in Proceedings of the symposium in foundations of mathematics, infinitistic methods, Pergamon Press, New York, 1961, pp. 303–321.Google Scholar