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Generalized Martin’s Axiom and Souslin’s hypothesis for higher cardinals

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Abstract

We consider different generalizations of Martin’s Axiom to higher cardinals. For ℵ1, assuming CH+2ℵ 1>ℵ2+□ℵ1 we show that a generalized Martin’s Axiom considered by Baumgartner settles the ℵ2 Souslin Hypothesis ... the wrong way. We further show that, assuming CH+2ℵ 1>ℵ2, a strengthening of this axiom implies □ℵ 1. Finally, we show that a seemingly innocuous further strengthening is inconsistent with CH+2ℵ 1>ℵ2.

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References

  1. K. Devlin,An alternative to Martin's Axiom, inSet Theory and Hierarchy Theory (Proc. 2nd Conf. Bierutowice '75), Lecture Notes in Math.547, Springer, 1976, pp. 65–76.

  2. J. Gregory,Higher Souslin trees, J. Symb. Logic41 (1976), 663–671.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. B. Jensen,The fine structure of the constructible hierarchy, Ann. Math. Log.4 (1972), 229–308.

    Article  MATH  Google Scholar 

  4. R. Laver and S. Shelah,The ℵ 2-Souslin-Hypothesis Trans. Am. Math. Soc.264 (1981), 411–418.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Shelah,A weak generalization of MAto higher cardinals, Isr. J. Math.30 (1978), 297–306.

    Article  MATH  Google Scholar 

  6. S. Shelah and L. Stanley, S-Forcing, I. A. “black-box” theorem for morasses, with applications to super-Souslin trees, Isr. J. Math.43 (1982), 185–224.

    MATH  MathSciNet  Google Scholar 

  7. L. Stanley, “L-Like” Models of Set Theory: Forcing, Combinatorial Principles and Morasses, Ph.D. Thesis, Univ. of Calif., Berkeley, 1977.

    Google Scholar 

  8. F. Tall,Some consequences of a generalized Martin's Axiom, to appear.

  9. D. Velleman, Ph.D. Thesis, University of Wisconsin, Madison, 1980.

  10. D. Velleman, Lecture Notes, Nserc Summer Meeting in Set Theory and Set-Theoretic Topology, University of Toronto, August, 1980.

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Author notes

  1. L. Stanley

    Present address: Department of Mathematics, Dartmouth College, 03755, Hanover, NH, USA

Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    S. Shelah & L. Stanley

  2. Maths Pures-Les Cezeaux, Université de Clermont II, 63170, Aubiere, France

    S. Shelah & L. Stanley

Authors
  1. S. Shelah
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  2. L. Stanley
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Additional information

This author thanks the US-Israel Binational Science Foundation for partial support of this research.

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Shelah, S., Stanley, L. Generalized Martin’s Axiom and Souslin’s hypothesis for higher cardinals. Israel J. Math. 43, 225–236 (1982). https://doi.org/10.1007/BF02761943

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  • DOI: https://doi.org/10.1007/BF02761943

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