We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-ℵ₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprāns and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-ℵ₁ property of a partial ordering while preserving its ccc-ness.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-2-6,
author = {Joan Bagaria and Saharon Shelah},
title = {On partial orderings having precalibre-1 and fragments of Martin's axiom},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {181-197},
zbl = {06498826},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-2-6}
}
Joan Bagaria; Saharon Shelah. On partial orderings having precalibre-ℵ₁ and fragments of Martin's axiom. Fundamenta Mathematicae, Tome 233 (2016) pp. 181-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-2-6/