Aronszajn trees, square principles, and stationary reflection

Chris Lambie-Hanson

doi:10.1002/malq.201600040

SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

Journal of Symbolic Logic ◽

10.1017/jsl.2018.56 ◽

2018 ◽

Vol 83 (04) ◽

pp. 1512-1538 ◽

Cited By ~ 2

Author(s):

CHRIS LAMBIE-HANSON ◽

PHILIPP LÜCKE

Keyword(s):

Partial Order ◽

Set Theory ◽

Consistency Strength ◽

Chain Conditions ◽

Regular Cardinals ◽

Aronszajn Tree ◽

Square Principles ◽

Image Position ◽

Aronszajn Trees ◽

Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

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Simultaneous stationary reflection and square sequences

Journal of Mathematical Logic ◽

10.1142/s0219061317500106 ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750010 ◽

Cited By ~ 4

Author(s):

Yair Hayut ◽

Chris Lambie-Hanson

Keyword(s):

Stationary Reflection ◽

Square Principles ◽

Square Sequences ◽

The Relationship ◽

Weak Square

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

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Strong Chang’s Conjecture, Semi-Stationary Reflection, the Strong Tree Property and two-cardinal square principles

Fundamenta Mathematicae ◽

10.4064/fm257-5-2016 ◽

2017 ◽

Vol 236 (3) ◽

pp. 247-262 ◽

Cited By ~ 1

Author(s):

Víctor Torres-Pérez ◽

Liuzhen Wu

Keyword(s):

Tree Property ◽

Stationary Reflection ◽

Chang’S Conjecture ◽

Square Principles ◽

Chang's Conjecture

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Rado’s Conjecture and its Baire version

Journal of Mathematical Logic ◽

10.1142/s0219061319500156 ◽

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.

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Knaster and friends II: The C-sequence number

Journal of Mathematical Logic ◽

10.1142/s0219061321500021 ◽

2020 ◽

Vol 21 (01) ◽

pp. 2150002

Author(s):

Chris Lambie-Hanson ◽

Assaf Rinot

Keyword(s):

General Formula ◽

Large Cardinals ◽

Weakly Compact ◽

Stationary Reflection ◽

Sequence Number ◽

Uncountable Cardinal ◽

Square Principles ◽

Cardinal Characteristic ◽

Independence Results

Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [Formula: see text] and independence results about the [Formula: see text]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [Formula: see text]-sequence spectrum and uncover some tight connections between the [Formula: see text]-sequence spectrum and the strong coloring principle [Formula: see text], introduced in Part I of this series.

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