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The Journal of Symbolic Logic The Journal of Symbolic Logic

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Semiproper forcing axiom implies Martin maximum but not PFA+

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Saharon Shelah*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
*
Institute of Mathematics, The Hebrew University, Jerusalem, Israel
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Abstract

We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA+ or even PFA+ (using the consistency of a large cardinal).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 2 , June 1987 , pp. 360 - 367
Copyright
Copyright © Association for Symbolic Logic 1987

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References

[1]Baumgartner, J., Applications of the proper forcing axiom, Handbook of set-theoretic topology (Kunen, K. and Vaughan, J. E., editors), North-Holland, Amsterdam, 1984, pp. 913–959.CrossRefGoogle Scholar
[2]Foreman, M., Magidor, M. and Shelah, S., Martin maximum, saturated ideals and nonregular ultrafilters. I, Annals of Mathematics (to appear).Google Scholar
[3]Laver, R., Making the supercompactness of κ: indestructible under κ-directed closed forcing, Israel Journal of Mathematics, vol. 29 (1978), pp. 385–388.CrossRefGoogle Scholar
[4]Shelah, S., Iterated forcing and changing cofinalities, Israel Journal of Mathematics, vol. 40 (1981), pp. 1–32.CrossRefGoogle Scholar
[5]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
[6]Todorčević, S., Forcing positive partition relations, Transactions of the American Mathematical Society, vol. 280 (1983), pp. 703–720.CrossRefGoogle Scholar
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