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Bounded Martin's Maximum, weak Erdӧs cardinals, and ψAc

Published online by Cambridge University Press:  12 March 2014

David Asperó  and
Philip D. Welch
David Asperó
Affiliation:
Institut Für Formale Logik, Universität Wien, Währingerstr. 25, A-1090 Wien, Austria, E-mail: [email protected]
Philip D. Welch
Affiliation:
Institut Für Formale Logik, Universität Wien, Währingerstr. 25, A-1090 Wien, Austria, E-mail: [email protected]
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Abstract

We prove that a form of the Erdӧs property (consistent with V = L[Hω2] and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle ψAC holds, and therefore . We also prove that ψAC implies that every function f: ω1 → ω1 is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum fails.

Keywords

Bounded Martin's MaximumψACbounding by canonical functionsErdӧs cardinals
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 1141 - 1152
Copyright
Copyright © Association for Symbolic Logic 2002

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References

REFERENCES

[Al]Asperó, David, Boundedforcing axioms and the continuum, Ph.D. thesis, University of Barcelona, 2000.Google Scholar
[A2]Asperó, David, Bounded forcing axioms and the size of the continuum, submitted.Google Scholar
[A-B]Asperó, David and Bagaria, Joan, Bounded forcing axioms and the continuum, Annals of Pure and Applied Logic, vol. 109 (2001), pp. 179–203.CrossRefGoogle Scholar
[B]Bagaria, Joan, Bounded forcing axioms as principles of generic absoluteness, Archive for Mathematical Logic, vol. 39 (2000), pp. 393–401.CrossRefGoogle Scholar
[Ba-G]Baumgartner, James and Galvin, Fred, Generalized Erdӧs cardinals and 0#, Annals of Mathematical Logic, vol. 15 (1978), pp. 289–313.CrossRefGoogle Scholar
[D-Do]Deiser, Oliver and Donder, Hans-Dieter, Canonical Functions, non-regular ultrafilters and Ulam's problem on ω1, submitted.Google Scholar
[Do-K]Donder, Hans-Dieter and Koepke, Peter, On the consistency strength of ‘accessible’ Jo´nsson cardinals, Annals of Mathematical Logic, vol. 25 (1983), pp. 233–261.Google Scholar
[Do-L]Donder, Hans-Dieter and Levinski, Jean-Pierre, Some principles related to Chang's Conjecture, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 39–101.CrossRefGoogle Scholar
[F-J]Feng, Qi and Jech, Thomas, Projective stationary sets and strong reflection principles, The Journal of the London Mathematical Society, vol. 58 (1998), pp. 271–283.CrossRefGoogle Scholar
[Fo-M-S]Foreman, Matthew, Magidor, Menachem, and Shelah, Saharon, Martin's Maximum, saturated ideals, and non–regular ultrafilters. Part I, Annals of Mathematics, vol. 127 (1988), pp. 1–47.CrossRefGoogle Scholar
[G-S]Goldstern, Martin and Shelah, Saharon, The bounded proper forcing axiom, this Journal, vol. 60 (1995), pp. 58–73.Google Scholar
[J]Jech, Thomas, Set theory. Second corrected edition, Perspectives in mathematical logic, Springer, Berlin-Heidelberg-New York, 1997.Google Scholar
[Mi]Miyamoto, Tadatoshi, A note on a weak segment of PFA, Proceedings of the sixth asian logic conference (Beijing, 1996) (River Edge, NJ), World Sci. Publishing, 1998, pp. 175–197.CrossRefGoogle Scholar
[S]Shelah, Saharon, Semiproperforcing axiom implies Martin maximum but not PFA+, this Journal, vol. 52 (1987), pp. 360–367.Google Scholar
[W]Welch, Philip, On unfoldable cardinals, ω-closed cardinals, and the beginning of the inner model hierarchy, submitted.Google Scholar
[Wo]Woodin, Hugh, The axiom of determinacy, forcing axioms, and the nonstationary ideal, Series in logic and its applications, number 1, De Gruyter, Berlin, New York, 1999.Google Scholar