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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
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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.

Information

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

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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