Hostname: page-component-7b9c58cd5d-hpxsc Total loading time: 0 Render date: 2025-03-14T16:04:17.543Z Has data issue: false hasContentIssue false
  • English
  • Français

On strongly minimal sets

Published online by Cambridge University Press:  12 March 2014

J. T. Baldwin  and
A. H. Lachlan
J. T. Baldwin
Affiliation:
Simon Fraser University, Burnaby 2, British Columbia
A. H. Lachlan
Affiliation:
Simon Fraser University, Burnaby 2, British Columbia
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 1 , March 1971 , pp. 79 - 96
Copyright
Copyright © Association for Symbolic Logic 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Keisler, H. J., Ultraproducts which are not saturated, this Journal, vol. 32 (1967), pp. 23–46.Google Scholar
[2]Keisler, H. J., Some model theoretic results for ω-logic, Israel Journal of Mathematics, vol. 4 (1966), pp. 249–261.CrossRefGoogle Scholar
[3]Marsh, W. E., On ω1-categorical but not ω-categorical theories, Doctoral Dissertation, Dartmouth College, 1966.Google Scholar
[4]Morley, M., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–518.CrossRefGoogle Scholar
[5]Morley, M., Countable models of ℵ1-categorical theories, Israel Journal of Mathematics, vol. 5 (1967), pp. 65–72.CrossRefGoogle Scholar
[6]Morley, M. and Vaught, R. L., Homogeneous universal models, Mathematica Scandinavica, vol. II (1962), pp. 37–57.CrossRefGoogle Scholar
[7]Park, D. M. R., Set theoretic constructions in model theory, Doctoral Dissertation, Massachusetts Institute of Technology, 1964.Google Scholar
[8]Shoenfield, J. R., Mathematical logic, Addison-Wesley, 1967.Google Scholar
[9]Vaught, R. L., Denumerable models of complete theories, Proceedings of the Symposium on Foundations of Mathematics: Infinitistic methods, Pergamon Press, New York, 1961, pp. 303–321.Google Scholar