On partial orderings having precalibre-$\aleph _1$ and fragments of Martin's axiom
Tom 232 / 2016
We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph _1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $\sigma $-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 $\sigma $-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprāns and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph _1$ property of a partial ordering while preserving its ccc-ness.