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CHAIN CONDITIONS OF PRODUCTS, AND WEAKLY COMPACT CARDINALS

Published online by Cambridge University Press:  24 October 2014

ASSAF RINOT
ASSAF RINOT*
Affiliation:
DEPARTMENT OF MATHEMATICS, BAR-ILAN UNIVERSITY, RAMAT-GAN 52900, ISRAEL, URL: http://www.assafrinot.comE-mail: [email protected]
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Abstract

The history of productivity of the κ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal $\kappa > \aleph _1 {\rm{,}}$ the principle □(k) is equivalent to the existence of a certain strong coloring $c\,:\,[k]^2 \, \to $k for which the family of fibers ${\cal T}\left( c \right)$ is a nonspecial κ-Aronszajn tree.

The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ-chain condition is productive for a given regular cardinal $\kappa > \aleph _1 {\rm{,}}$ then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ-chain condition is productive.

Keywords

Aronszajn treechain conditionsquarewalks on ordinalsweakly compact cardinal
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 3 , September 2014 , pp. 293 - 314
Copyright
Copyright © The Association for Symbolic Logic 2014 

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