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Extended ultrapowers and the Vopěnka-Hrbáček theorem without choice

Published online by Cambridge University Press:  12 March 2014

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

We generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopěnka-Hrbáček theorem from ZF + DC only (the proof of Vopěnka and Hrbáček used the full axiom of choice): if there exists a strongly compact cardinal, then the universe is not constructible from a set. The same method shows that, in L[2ω], there cannot exist a θ-compact cardinal less than θ (where θ is the least cardinal onto which the continuum cannot be mapped); a similar result can be proven for other models of the form L[A]. The result for L[2ω] is of particular interest in connection with the axiom of determinacy. The extended ultrapower construction of this paper is an improved version of the author's earlier pseudo-ultrapower method, making use of forcing rather than the omitting types theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 2 , June 1991 , pp. 592 - 607
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

[B1]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
[DH]DiPrisco, C. A. and Henle, J. M., On the compactness of ℵ1 and ℵ2, this Journal, vol. 43 (1978), pp. 394–401.Google Scholar
[FJ]Friedman, H. and Jensen, R., Note on admissible ordinals, The syntax and semantics of infinitary languages (Barwise, J., editor), Lecture Notes in Mathematics, vol. 72, Springer-Verlag, Berlin, 1968, pp. 77–79.CrossRefGoogle 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
[He1]Henle, J. M., Spector forcing, this Journal, vol. 49 (1984), pp. 542–554.Google Scholar
[He2]Henle, J. M., An extravagant partition relation for a model of arithmetic, Axiomatic set theory (Baumgartner, J. E.et al., editors), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 109–113.CrossRefGoogle Scholar
[J]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[KKMW]Kechris, A. S., Kleinberg, E. M., Moschovakis, Y. N., and Woodin, W. H., The axiom of determinacy, strong partition relations and non-singular measures, Cabal seminar 77–79 (Kechris, A. S.et al., editors), Lecture Notes in Mathematics, vol. 839, Springer-Verlag, Berlin, 1981, pp. 75–99.CrossRefGoogle Scholar
[Ke1]Keisler, H. J., Model theory for infinitary logic: logic with countable conjunctions and finite quantifiers, North-Holland, Amsterdam, 1971.Google Scholar
[Ke2]Keisler, H. J., Forcing and the omitting types theorem, Studies in model theory (Morley, M. D., editor), MAA Studies in Mathematics, vol. 8, Mathematical Association of America, Buffalo, New York, 1973, pp. 96–133.Google Scholar
[Ku]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179–227.CrossRefGoogle Scholar
[Ł]Łoś, J., Quelques remarques, théorèmes et problèmes sur les classes définissables d'algèbres, Mathematical interpretation of formal systems (Skolem, T.et al., editors), North-Holland, Amsterdam, 1955, pp. 98–113.CrossRefGoogle Scholar
[Mag]Magidor, M., How large is the first strongly compact cardinal? Annals of Mathematical Logic, vol. 10(1976), pp. 33–57.CrossRefGoogle Scholar
[Mar]Martin, D. A., Unpublished.Google Scholar
[Mat]Mathias, A. R. D., Happy families, Annals of Mathematical Logic, vol. 12(1977), pp. 59–111.CrossRefGoogle Scholar
[Sa]Sacks, G. E., Countable ordinals and hyperdegrees, Advances in Mathematics, vol. 20 (1976), pp. 213–262.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
[So1]Solovay, R. M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, ser. 2, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar
[So2]Solovay, R. M., Strongly compact cardinals and the G. C. H., Proceedings of the Tarski symposium (Henkin, L. et al., editors), Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, Rhode Island, 1974, pp. 365–372.CrossRefGoogle Scholar
[So3]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
[SRK]Solovay, R. M., Reinhardt, W. N., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar
[Sp1]Spector, M., A measurable cardinal with a nonwellfounded ultrapower, this Journal, vol. 45 (1980), pp. 623–628.Google Scholar
[Sp2]Spector, M., Ultrapowers without the axiom of choice, this Journal, vol. 53 (1988), pp. 1208–1219.Google Scholar
[VH]Vopěnka, P. and Hrbáček, K., On strongly measurable cardinals, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathematiques, Astronomiques et Physiques, vol. 14 (1966), pp. 587–591.Google Scholar
[W]Woodin, W. H., absoluteness and supercompact cardinals, photocopied notes, 1985.Google Scholar