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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 (hsakai@people.kobe-u.ac.jp)
Boban Veličković
Affiliation:
Institut de Mathematiques de Jussieu - Paris Rive Gauche, Université Paris Diderot, 75205 Paris Cedex 13, France (boban@math.univ-paris-diderot.fr)
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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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