Martin Axiom and the size of the continuum

Haim Judah and Andrzej Roslanowski

We investigate various Forcing Axioms. We show that ${\rmMA}_{\omega_1}(\mbox{ccc})$ implies ${\rm MA}_{\omega_1}(\sigma\mbox{-closed}*\sigma\mbox{-linked})$ and ${\rm MA}_{\omega_1}(\sigma\mbox{-closed}*\mbox{projective ccc})$. Using this we generalize a result of Shelah proving: ${\rmMA}_{\omega_1}(\mbox{ccc})$ implies that every $\Sigma^1_2$ ccc forcing notion adding an unbounded real adds a Cohen real.

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