Abstract
is stationary. In general, if X is any class of posets, the forcing axioms XFA and XFA + are obtained by replacing “semi-proper LY” by “9 E X” in the above definitions. Sometimes we denote these axioms by FA(X) and FA+(.X). Thus, for example, FA(ccc) is the familiar Martin’s Axiom, MAN,. The notion of semi-proper, as an extension of a previous notion of proper, was defined and extensively studied by Shelah in [Shl J. These forcing notions generalize ccc and countably closed forcing and can be iterated without collapsing N, . SPFA is implicit in [Shl] as an obvious strengthening of the Proper Forcing Axiom (PFA), previously formulated and proved consistent by Baumgartner (see [Bal, Del]). PFA had settled many combinatorial questions on the uncountable and SPFA was not
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