Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T18:30:09.622Z Has data issue: false hasContentIssue false
  • English
  • Français

AD and patterns of singular cardinals below Θ

Published online by Cambridge University Press:  12 March 2014

Arthur W. Apter
Arthur W. Apter*
Affiliation:
Department of Mathematics, Baruch College of CUNY, New York, New York 10010, E-mail: AWABB@CUNYVM.CUNY.EDU
Get access Rights & Permissions [Opens in a new window]

Abstract

Using Steel's recent result that assuming AD, in L[ℝ] below Θ, κ is regular iff κ is measurable, we mimic below Θ certain earlier results of Gitik. In particular, we construct via forcing a model in which all uncountable cardinals below Θ are singular and a model in which the only regular uncountable cardinal below Θ is ℵ1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 1 , March 1996 , pp. 225 - 235
Copyright
Copyright © Association for Symbolic Logic 1996

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]Apter, A., On a problem inspired by determinacy, Israel Journal of Mathematics, vol. 61 (1988), pp. 256–270.CrossRefGoogle Scholar
[2]Apter, A., Some new upper bounds in consistency strength for certain choiceless large cardinal patterns, Archive for Mathematical Logic, vol. 31 (1992), pp. 201–205.CrossRefGoogle Scholar
[3]Cummings, J. and Woodin, H., Generalised Prikry forcings, circulated manuscript of a forthcoming book.Google Scholar
[4]Gitik, M., All uncountable cardinals can be singular, Israel Journal of Mathematics, vol. 35 (1980), pp. 61–88.CrossRefGoogle Scholar
[5]Gitik, M., Regular cardinals in models of ZF, Transactions of the American Mathematical Society, vol. 290 (1985), pp. 41–68.Google Scholar
[6]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[7]Kanamori, A. and Magidor, M., The evolution of large cardinal axioms in set theory, Lecture notes in mathematics, no. 669, Springer-Verlag, Berlin and New York, 1978, pp. 99–275.Google Scholar
[8]Kechris, A., The axiom of determinacy implies dependent choices in L[ℝ], this Journal, vol. 49 (1984), pp. 255–267.Google Scholar
[9]Kechris, A., Determinacy and the structure of L[ℝ], Proceedings of symposia in pure mathematics, no. 42, American Mathematical Society, Providence, RI, 1985, pp. 271–283.Google Scholar
[10]Kechris, A., Kleinberg, E., Moschovakis, Y., and Woodin, H., The axiom of determinacy, strong partition properties, and non-singular measures, Cabal seminar '77–' 79 (Kechris, A., Martin, D. A., and Moschovakis, Y., editors), Lecture Notes in Mathematics, no. 839, Springer-Verlag, Berlin and New York, 1981, pp. 75–100.CrossRefGoogle Scholar
[11]Kechris, A., Martin, D. A., and Moschovakis, Y. (editors), Cabal seminar '77–'79, Lecture Notes in Mathematics, no. 839, Springer-Verlag, Berlin and New York, 1981.CrossRefGoogle Scholar
[12]Kleinberg, E., AD ⊢ the ℵn are Jonsson cardinals and ℵω is a Rowbottom cardinal, Annals of Mathematical Logic, vol. 12 (1977), pp. 229–248.CrossRefGoogle Scholar
[13]Kleinberg, E., Infinitary combinatorics and the axiom of determinateness, Lecture notes in mathematics, no. 612, Springer-Verlag, Berlin and New York, 1977.Google Scholar
[14]Lévy, A. and Solovay, R., Measurable cardinals and the continuum hypothesis, Israel Journal of Mathematics, vol. 5 (1967), pp. 234–248.CrossRefGoogle Scholar
[15]Martin, D. A. and Steel, J., Projective determinacy, Proceedings of the National Academy of Sciences of the U.S. A, vol. 85 (1988), pp. 6582–6586.CrossRefGoogle ScholarPubMed
[16]Martin, D. A. and Steel, J., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar
[17]Moschovakis, Y., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[18]Rowbottom, F., Some strong axioms of infinity incompatible with the axiom of constructibility, Annals of Mathematical Logic, vol. 3 (1971), pp. 1–44.CrossRefGoogle Scholar
[19]Solovay, R., Reinhardt, W., and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar
[20]Steel, J., paper 1, in preparation.Google Scholar
[21]Steel, J., paper 2, in preparation.Google Scholar