Fragments of Martin's Maximum in generic extensions

Abstract

Abstract We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly ( ω 1 + 1)‐game‐closed forcings. PFA can be destroyed by a strongly ( ω 1 + 1)‐game‐closed forcing but not by an ω 2 ‐closed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)