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Ultrapowers without the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Mitchell Spector
Mitchell Spector*
Affiliation:
Department of Software Engineering and Computer Science, Seattle University, Seattle, Washington 98122
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Abstract

A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this is arranged by means of an application of the omitting types theorem. The general theory of pseudo-ultrapowers is developed. Following that, we study supercompactness in the absence of choice, and we analyze pseudo-ultrapowers of models of the axiom of determinateness and various infinite exponent partition relations. Relationships between pseudo-ultrapowers and forcing are also discussed.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 4 , December 1988 , pp. 1208 - 1219
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCE

[Bl] Becker, H., AD and the supercompactness of ℵ1 , this Journal, vol. 46 (1981), pp. 822–842.Google Scholar
[B2] Becker, H., Determinacy implies that ℵ2 is supercompact, Israel Journal of Mathematics, vol. 40 (1981), pp. 229–234.CrossRefGoogle Scholar
[CK] Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1973.Google Scholar
[DH] Di Prisco, C. A. and Henle, J. M., On the compactness of ℵ1 and ℵ2 , this Journal, vol. 43 (1978), pp. 394–401.Google Scholar
[HaK] Harrington, L. A. and Kechris, A. S., On the determinacy of games on ordinals, Annals of Mathematical Logic, vol. 20 (1981), pp. 109–154.CrossRefGoogle Scholar
[Hel] Henle, J. M., Researches into the world of k → (k) κ , Annals of Mathematical Logic, vol. 17 (1979) , pp. 151–169.CrossRefGoogle Scholar
[He2] Henle, J. M., Magidor-like and Radin-like forcing, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 59–72.CrossRefGoogle Scholar
[He3] Henle, J. M., Spector forcing, this Journal, vol. 49 (1984), pp. 542–554.Google Scholar
[He4] Henle, J. M., An extravagant partition relation for a model of arithmetic, Axiomatic set theory, Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 109–113.CrossRefGoogle Scholar
[Ke] Keisler, H. Jerome, Model theory for infinitary logic: logic with countable conjunctions and finite quantifiers, North-Holland, Amsterdam, 1971.Google Scholar
[K1] Kleinberg, E. M., Infinitary combinatorics and the axiom of determinateness, Lecture Notes in Mathematics, vol. 612, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
[Sc] Scott, D., Measurable cardinals and constructive sets, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9 (1961), pp. 521–524.Google Scholar
[So] Solovay, R. M., The independence of DC from AD, Cabal Seminar 76–77 (Kechris, A. S. and Moschovakis, Y. N., editors), Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin, 1978, pp. 171–184.CrossRefGoogle Scholar
[Sp] Spector, M., A measurable cardinal with a nonwellfounded ultrapower, this Journal, vol. 45 (1980), pp. 623–628.Google Scholar
[SpSt] Spector, M. and Stern, A. S., in preparation.Google Scholar