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Uniqueness and characterization of prime models over sets for totally transcendental first-order theories

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
Affiliation:
University of California, Los Angeles, Los Angeles, California 90024

Extract

If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.

The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this follows

If T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)

By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.

We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1972

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References

[1]Chang, C. C. and Keisler, H. J., Model theory, Appleton-Century-Crofts, 1971.Google Scholar
[2]Harnik, V. and Ressayre, J. P., Prime extensions and categoricity in power, Israel Journal of Math., to appear.Google Scholar
[3]Morley, M. D., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514538.CrossRefGoogle Scholar
[4]Morley, M. D., Countable models of ℵ1-categorical theories, Israel Journal of Math., vol. 5 (1967), pp. 6572.CrossRefGoogle Scholar
[5]Morley, M. D. and Vaught, R. L., Homogeneous universal models, Mathematica Scandinavica, vol. II (1962), pp. 3757.CrossRefGoogle Scholar
[6]Shelah, S., Stability, the f.c.p., and super stability; model theoretic properties of formulas in first order theory, Annals of Mathematical Logic, vol. 3 (1971).CrossRefGoogle Scholar
[7]Shelah, S., Some unconnected results in model theory, Notices of the American Mathematical Society, vol. 18 (1971), p. 563, Abstract #7IT-E30.Google Scholar
[8]Blum, L., Generalized algebraic structures: model theoretic approach Ph.D. Thesis, Massachusetts Institute of Technology, 1968.Google Scholar
[9]Vaught, R. L., Denumerable models of complete theories, Proceedings of the Symposium on Foundation of Mathematics, Warsaw, 7959, New York, Oxford, London, Paris and Warsaw, 1961, pp. 303321.Google Scholar
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