Separating stationary reflection principles

2000 ◽  
Vol 65 (1) ◽  
pp. 247-258 ◽  
Author(s):  
Paul Larson
Keyword(s):  
Stationary Reflection ◽  
Reflection Principles

AbstractWe present a variety of (ω, ∞)-distributive forcings which when applied to models of Martin's Maximum separate certain well known reflection principles. In particular, we do this for the reflection principles SR, SRα (α ≤ ω1), and SRP.

scholarly journals Stationary reflection principles and two cardinal tree properties

2013 ◽  
Vol 14 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Hiroshi Sakai ◽  
Boban Veličković
Keyword(s):  
Weak Form ◽  
Reflection Principle ◽  
Singular Cardinal ◽  
Stationary Reflection ◽  
Strong Compactness ◽  
Singular Cardinal Hypothesis ◽  
Martin’S Axiom ◽  
Reflection Principles ◽  
Stationary Set ◽  
Weak Square

AbstractWe 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.

scholarly journals Rado’s Conjecture and its Baire version

2019 ◽  
Vol 20 (01) ◽  
pp. 1950015
Author(s):  
Jing Zhang
Keyword(s):  
Reflection Principle ◽  
Stationary Reflection ◽  
Forcing Axioms ◽  
Martin’S Axiom ◽  
Partition Relations ◽  
Consistency Results ◽  
Square Principles ◽  
Reflection Principles ◽  
Weak Square ◽  
Forcing Axiom

Rado’s Conjecture is a compactness/reflection principle that says any nonspecial tree of height [Formula: see text] has a nonspecial subtree of size [Formula: see text]. Though incompatible with Martin’s Axiom, Rado’s Conjecture turns out to have many interesting consequences that are also implied by certain forcing axioms. In this paper, we obtain consistency results concerning Rado’s Conjecture and its Baire version. In particular, we show that a fragment of [Formula: see text], which is the forcing axiom for Baire Indestructibly Proper forcings, is compatible with the Baire Rado’s Conjecture. As a corollary, the Baire Rado’s Conjecture does not imply Rado’s Conjecture. Then we discuss the strength and limitations of the Baire Rado’s Conjecture regarding its interaction with stationary reflection principles and some families of weak square principles. Finally, we investigate the influence of Rado’s Conjecture on some polarized partition relations.

scholarly journals Semistationary and stationary reflection

2008 ◽  
Vol 73 (1) ◽  
pp. 181-192 ◽  
Author(s):  
Hiroshi Sakai
Keyword(s):  
Reflection Principle ◽  
Stationary Subset ◽  
Stationary Reflection ◽  
Regular Cardinals ◽  
Reflection Principles ◽  
The Relationship

AbstractWe study the relationship between the semistationary reflection principle and stationary reflection principles. We show that for all regular cardinals λ ≥ ω2 the semistationary reflection principle in the space [λ]ω implies that every stationary subset of ≔ {α ∈ λ ∣ cf(α) = ω} reflects. We also show that for all cardinals λ ≥ ω3 the semistationary reflection principle in [λ]ω does not imply the stationary reflection principle in [λ]ω.

scholarly journals Stationary reflection and level by level equivalence

2009 ◽  
Vol 115 (1) ◽  
pp. 113-128
Author(s):  
Arthur W. Apter
Keyword(s):  
Stationary Reflection
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