Abstract
The history of productivity of the k-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 k > the principle □() is equivalent to the existence of a certain strong coloring c : [∗]2 k for which the family of fibers T(c) is a nonspecial K-Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the /c-chain condition is productive for a given regular cardinal ∗ > N |. then k is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if k is a weakly compact cardinal, then the /c-chain condition is productive.
Original language | English |
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Pages (from-to) | 293-314 |
Number of pages | 22 |
Journal | Bulletin of Symbolic Logic |
Volume | 20 |
Issue number | 3 |
DOIs | |
State | Published - 1 Sep 2014 |
Bibliographical note
Publisher Copyright:©2014, Association for Symbolic Logic.
Keywords
- Aronszajn tree
- Chain condition
- Square
- Walks on ordinals
- Weakly compact cardinal