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Limit ultraproducts

Published online by Cambridge University Press:  12 March 2014

H. Jerome Keisler
H. Jerome Keisler*
Affiliation:
University of Wisconsin
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Extract

This paper is a sequel to our earlier paper, “Limit Ultrapowers”, [6]. In that paper we introduced the limit ultrapower construction and proved that is isomorphic to a limit ultrapower of if and only if every PCΔ class which contains also contains . In Section 1 of this paper we introduce the more general limit ultraproduct construction, and in Section 2 we prove that, for any class K of relational systems, a relational system is isomorphic to a limit ultraproduct of members of K if and only if every PCΔ class which includes K also contains . As a consequence, the property of K being an intersection of PCΔ classes is characterized purely set-theoretically by the property of K being closed under isomorphisms and limit ultraproducts.

In Section 3 we apply limit ultraproducts to obtain model-theoretic conditions equivalent to the set-theoretic condition that every α-complete ultrafilter is γ+-complete. The first result, Theorem 3.7, was announced in the abstract [8], and it is also closely related to a result which was stated without proof in [10], namely Theorem 2 of that paper.

In Sections 4 and 5 we apply our results in order to improve a theorem of Craig in [2]. Craig considered the logic L(Q), where Q is a set of cardinals, obtained from ordinary first order logic by adding for each α ϵ Q the quantifier “there exist at least α”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 30 , Issue 2 , June 1965 , pp. 212 - 234
Copyright
Copyright © Association for Symbolic Logic 1965

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References

REFERENCES

[1]Chang, C. C. and Keisler, H. J., Continuous Model Theory, Princeton, Annals of Mathematics Studies, to appear.CrossRefGoogle Scholar
[2]Craig, W., Relative characterizability and generalized existential quantifiers, Notices American Mathematical Society, vol. 9 (1962), p. 153, Abstract 62T-96.Google Scholar
[3]Craig, W. and Hanf, W., On relative characterizability in a language, Notices American Mathematical Society, vol. 9 (1962), pp. 152–153, Abstract 62T-95.Google Scholar
[4]Frayne, T., Morel, A. C. and Scott, D., Reduced direct products, Fundamenta Mathematicae, vol. 51 (1962), pp. 195–228.CrossRefGoogle Scholar
[5]Fuhrken, G., On generalized quantifiers (abstract), Notices American Mathematical Society, vol. 9 (1962), p. 132.Google Scholar
[6]Keisler, H. J., Limit ultrapowers, Transactions American Mathematical Society, vol. 107 (1963), pp. 382–408.CrossRefGoogle Scholar
[7]Keisler, H. J., Limit reduced products, Notices American Mathematical Society, vol. 6 (1959), p. 520 (Abstract 559-137).Google Scholar
[8]Keisler, H. J., The equivalence of certain problems in set theory with problems in the theory of models, Notices American Mathematical Society, vol. 9 (1962), pp. 339–340, Abstract 62T-257.Google Scholar
[9]Keisler, H. J., Intersections of PCΔ classes, Notices American Mathematical Society, vol. 10 (1963), p. 137, Abstract 63T-48. Errata, Notices American Mathematical Society, vol. 10 (1963), p. 377.Google Scholar
[10]Keisler, H. J., Some applications of the theory of models to set theory. An article in Logic, Methodology, and the Philosophy of Science, pp. 80–86, Stanford, 1962.Google Scholar
[11]Keisler, H. J., On cardinalities of ultrapowers, Bulletin American Mathematical Society, vol. 70 (1964), pp. 644–647.CrossRefGoogle Scholar
[12]Keisler, H. J. and Tarski, A., From accessible to inaccessible cardinals, Fundamentia Mathematicae, vol. 53 (1964), pp. 225–308.CrossRefGoogle Scholar
[13]Kochen, S., Ultraproducts in the theory of models, Annals of Mathematics, vol. 74 (1961), pp. 221–261.CrossRefGoogle Scholar
[14]Morley, M. and Vaught, R., Homogeneous universal models, Mathematica Scandinavica, vol. 11 (1962), pp. 37–57.CrossRefGoogle Scholar
[15]Mostowski, A., On a generalization of quantifiers, Fundamenta Mathematicae, vol. 44 (1957), pp. 12–36.CrossRefGoogle Scholar
[16]Rabin, M. O., Arithmetical extensions with prescribed cardinality, Indagationes Mathematicae, vol. 21 (1959), pp. 439–446.CrossRefGoogle Scholar
[17]Tarski, A., Quelques theorems sur les alephs, Fundamenta Mathematicae, vol. 7 (1925), pp. 1–14.CrossRefGoogle Scholar
[18]Tarski, A., Contributions to the theory of models, I and II, Indagationes Mathematicae, vol. 16 (1954), pp. 572–588.CrossRefGoogle Scholar
[19]Tarski, A., Some model-theoretic results concerning weak second-order logic, Notices American Mathematical Society, vol. 6 (1958), p. 673, Abstract 550-6.Google Scholar