Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-22T14:16:20.940Z Has data issue: false hasContentIssue false
  • English
  • Français

CHANG’S CONJECTURE, GENERIC ELEMENTARY EMBEDDINGS AND INNER MODELS FOR HUGE CARDINALS

Published online by Cambridge University Press:  15 September 2015

MATTHEW FOREMAN
MATTHEW FOREMAN*
Affiliation:
MATHEMATICS DEPARTMENT UC IRVINE IRVINE, CA, 92697-3875, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a natural principle Strong Chang Reflection strengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show that decisive ideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found in [10].1,2

Keywords

Chang’s Conjectureinner modelshuge cardinals
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 21 , Issue 3 , September 2015 , pp. 251 - 269
Copyright
Copyright © The Association for Symbolic Logic 2015 

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

Chang, C. C. and Keisler, H. J., Model Theory, Third edition, Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam, 1990, pp. xvi+650.Google Scholar
Cummings, J., Collapsing successors of singulars. Proceedings of the American Mathematical Society, vol. 125 (1997), no. 9, pp. 2703–2709.CrossRefGoogle Scholar
Cummings, J., Foreman, M., and Magidor, M., Canonical structure in the universe of set theory. I. Annals of Pure and Applied Logic, vol. 129 (2004), no. 1–3, pp. 211–243.CrossRefGoogle Scholar
Devlin, Keith, Aspects of Constructibility, Lecture Notes in Mathematics, vol. 354, Springer-Verlag, Berlin–New York, 1973, pp. xii +240.CrossRefGoogle Scholar
Erdős, Paul, Hajnal, Andrs, Mt, Attila, and Rado, Richard, Combinatorial Set Theory: Partition Relations for Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, North-Holland, Amsterdam, 1984, p.347.Google Scholar
Foreman, M., Large cardinals and strong model theoretic transfer properties. Transactions of the American Mathematical Society, vol. 272 (1982), no. 2, pp. 427–463.CrossRefGoogle Scholar
Foreman, M., More saturated ideals, Cabal Seminar 79–81, Springer, Berlin, 1983, pp. 1–27.Google Scholar
Foreman, M., Calculating quotient algebras of generic embeddings. Israel Journal of Mathematics, vol. 193 (2013), no. 1, pp. 309–341.CrossRefGoogle Scholar
Foreman, M., Ideals and generic elementary embeddings, Handbook of Set Theory, Vols. 1, 2, 3, pp. 885–1147, Springer, Dordrecht, 2010.CrossRefGoogle Scholar
Foreman, M., Smoke and Mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals. Advances in Mathematics, vol. 222 (2009), no. 2, pp. 565–595.CrossRefGoogle Scholar
Foreman, M., Stationary sets, Chang’s conjecture and partition theory, Set Theory (Piscataway, NJ, 1999), DIMACS Series. Discrete Mathematics and Theoretical Computer Science, vol. 58, American Mathematical Society, Providence, RI, 2002, pp. 73–94.CrossRefGoogle Scholar
Foreman, M. and Magidor, M., Large cardinals and definable counterexamples to the continuum hypothesis. Annals of Pure and Applied Logic, vol. 76 (1995), no. 1, pp. 47–97.CrossRefGoogle Scholar
Foreman, M., Magidor, M., and Shelah, S., Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Annals of Mathematics (2), vol. 127 (1988), no. 1, pp. 1–47.CrossRefGoogle Scholar
Gitik, M., On generic elementary embeddings. Journal of Symbolic Logic, vol. 54 (1989), no. 3, pp. 700–707.CrossRefGoogle Scholar
Jech, Thomas J., Some combinatorial problems concerning uncountable cardinals. Annals of Mathematical Logic, vol. 5 (1972/73), pp. 165–198.CrossRefGoogle Scholar
Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Higher Set Theory (Proceedings of the Conference Mathematics, Forschungsinst., Oberwolfach, 1977), Lecture Notes in Mathematics, vol. 669, Springer, Berlin, 1978, pp. 99–275.Google Scholar
Kunen, K., Saturated ideals. Journal of Symbolic Logic, vol. 43 (1978), no. 1, pp. 65–76.CrossRefGoogle Scholar
Levinski, Jean-Pierre, Magidor, Menachem, and Shelah, Saharon, Chang’s conjecture for אּω. Israel Journal of Mathematics, vol. 69 (1990), no. 2, pp. 161–172.CrossRefGoogle Scholar
Lindström, Per, On extensions of elementary logic. Theoria, vol. 35 (1969), pp. 1–11.CrossRefGoogle Scholar
Sargsyan, Grigor, Descriptive inner model theory, this Bulletin, vol. 19 (2013), no.1, pp. 1–55 .CrossRefGoogle Scholar
Schimmerling, Ernest, A core model toolbox and guide, Handbook of Set Theory, Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1685–1751.CrossRefGoogle Scholar
Schindler, R. and Steel, J., The core model induction, unpublished manuscript available at: https://wwwmath.uni-muenster.de/u/rds/core_model_induction.pdf.Google Scholar
Schmerl, J. H., Chapter V: Transfer theorems and their applications to logics, Model-Theoretic Logics, Perspectives in Logic, vol. 8, Springer-Verlag, New York, 1985, pp. 177–209.Google Scholar
Silver, Jack, The consistency of the GCH with the existence of a measurable cardinal, Axiomatic Set Theory (Proceedings of Symposia in Pure Mathematics, Vol. XIII, Part I, University of California, Los Angeles, California, 1967), American Mathematical Society, Providence, R.I, pp. 391–395.Google Scholar
Solovay, R., Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium (Proceedings of Symposia in Pure Mathematics, Vol. XXV, Univeristy of California, Berkeley, California, 1971), American Mathematical Society, Providence, R.I., 1974, pp. 365–372.Google Scholar
Vaught, R. L., The Löwenheim-Skolem theorem, Logic, Methodology and Philosophy of Science (Proceedings of the 1964 International Congress), North-Holland, Amsterdam, 1965, pp. 81–89.Google Scholar