On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom

Joan Bagaria; Saharon Shelah

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arXiv · 1502.05500

On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom

Joan Bagaria·Saharon Shelah

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Abstract

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 answers 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. Steprans 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.

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Joan Bagaria, Saharon Shelah. 2015-02-19. On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom. https://arxiv.org/abs/1502.05500

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