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SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

Published online by Cambridge University Press:  21 December 2018

CHRIS LAMBIE-HANSON  and
PHILIPP LÜCKE
CHRIS LAMBIE-HANSON
Affiliation:
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS VIRGINIA COMMONWEALTH UNIVERSITY 1015 FLOYD AVENUE, RICHMOND VIRGINIA 23284, USAE-mail: cblambiehanso@vcu.eduURL: http://people.vcu.edu/∼cblambiehanso/
PHILIPP LÜCKE
Affiliation:
MATHEMATISCHES INSTITUT UNIVERSITÄT BONN ENDENICHER ALLEE 60, 53115 BONN, GERMANYE-mail: pluecke@math.uni-bonn.deURL: http://www.math.uni-bonn.de/people/pluecke/
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Abstract

With 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 $\square \left( \kappa \right)$.

Keywords

Primary 03E05Secondary 03E3503E55square principlesascent pathsspecial treesproductivity of chain conditionsKnaster propertylayered posetswalks on ordinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1512 - 1538
Copyright
Copyright © The Association for Symbolic Logic 2018 

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