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The number of countable models

  • Michael Morley (a1)

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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.

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[1]Friedman, H., On the ordinals in models of set theory, preprint.
[2]Kuratowski, K., Topology, Academic Press, New York, 1966.
[3]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers in Theory of models, North-Holland, Amsterdam, 1965, pp. 329341.
[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. 303321.

The number of countable models

  • Michael Morley (a1)

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