Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-25T03:47:18.101Z Has data issue: false hasContentIssue false

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

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

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