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Stationary reflection principles and two cardinal tree properties

Part of: Set theory

Published online by Cambridge University Press:  01 November 2013

Hiroshi Sakai  and
Boban Veličković
Hiroshi Sakai
Affiliation:
Department of Computer Science and Systems Engineering Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan ([email protected])
Boban Veličković
Affiliation:
Institut de Mathematiques de Jussieu - Paris Rive Gauche, Université Paris Diderot, 75205 Paris Cedex 13, France ([email protected])
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Abstract

We study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.

Keywords

stationary setsreflectiontree propertylarge cardinals

MSC classification

Secondary: 03E04: Ordered sets and their cofinalities; pcf theory 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom 03E55: Large cardinals 03E05: Other combinatorial set theory 03E65: Other hypotheses and axioms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 1 , January 2015 , pp. 69 - 85
Copyright
©Cambridge University Press 2013 

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References

Baumgartner, J. E., On the size of closed unbounded sets, Ann. Pure Appl. Logic 54 (3) (1991), 195–227.Google Scholar
Foreman, M., Magidor, M. and Shelah, S., Martin’s maximum, saturated ideals, and nonregular ultrafilters. I, Ann. of Math. (2) 127 (1) (1988), 1–47.Google Scholar
Jech, T. J., Some combinatorial problems concerning uncountable cardinals, Ann. Math. Logic 5 (1972/73), 165–198.Google Scholar
Kanamori, A., Large cardinals in set theory from their beginnings, in The higher infinite, 2nd edn, Springer Monographs in Mathematics (Springer, Berlin, 2003).Google Scholar
Krueger, J., Strong compactness and stationary sets, J. Symbolic Logic 70 (3) (2005), 767–777.Google Scholar
Magidor, M., Combinatorial characterization of supercompact cardinals, Proc. Amer. Math. Soc. 42 (1974), 279–285.Google Scholar
Sakai, H., Semistationary and stationary reflection, J. Symbolic Logic 73 (1) (2008), 181–192.Google Scholar
Shelah, S., Semiproper forcing axiom implies Martin maximum but not ${\mathrm{PFA} }^{+ } $ , J. Symbolic Logic 52 (2) (1987), 360–367.Google Scholar
Shelah, S., Cardinal arithmetic, Oxford Logic Guides, Volume 29 (The Clarendon Press Oxford University Press, New York, 1994), Oxford Science Publications.Google Scholar
Shelah, S., Proper and improper forcing, 2nd edn, Perspectives in Mathematical Logic (Springer, Berlin, 1998).Google Scholar
Silver, J., On the singular cardinals problem, in Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Volume 1, pp. 265–268 (Canad. Math. Congress, Montreal, Que, 1975).Google Scholar
Todorčević, S., Conjectures of Rado and Chang and cardinal arithmetic, in Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Volume 411, pp. 385–398 (Kluwer Acad. Publ, Dordrecht, 1993).Google Scholar
Veličković, B., Forcing axioms and stationary sets, Adv. Math. 94 (2) (1992), 56–284.Google Scholar
Weiss, C., Subtle and ineffable tree properties, PhD thesis, Ludwig Maximilians Universität, München (2010).Google Scholar
Weiss, C. and Viale, M., On the consistency strength of the proper forcing axiom, Adv. Math. 228 (5) (2011), 2672–2687.Google Scholar