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On strongly minimal sets

  • J. T. Baldwin (a1) and A. H. Lachlan (a1)
  • DOI:
  • Published online: 01 March 2014

The purpose of this paper is twofold. In §1 and §2 which are largely expository we develop the known theory of ℵ1-categoricity in terms of strongly minimal sets. In §3 we settle affirmatively Vaught's conjecture that a complete ℵ1-categorical theory has either just one or just ℵ0 countable models, and in §4 we present an example which serves to illustrate the ideas of §3.

As far as we know the only work published on strongly minimal sets is that of Marsh [3]. The present exposition goes beyond [3] in showing that any ℵ-categorical theory has a principal extension in which some formula is strongly minimal.

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[2]H. J. Keisler , Some model theoretic results for ω-logic, Israel Journal of Mathematics, vol. 4 (1966), pp. 249261.

[4]M. Morley , Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514518.

[5]M. Morley , Countable models of ℵ1-categorical theories, Israel Journal of Mathematics, vol. 5 (1967), pp. 6572.

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