The combinatorial principle ?(?) says that there is a coherent sequence of length ? that cannot be threaded. If ?=?+, then the related principle ?? implies ?(?). Let ?2 and X?. Assume both ?(?) and ?? fail. Then there is an inner model N with a proper class of strong cardinals such that XN. If, in addition, ?20 and n, then there is an inner model Mn(X) with n Woodin cardinals such that XMn(X). In particular, by Martin and Steel, Projective Determinacy holds. As a corollary to this and results of Todorcevic and Velickovic, the Proper Forcing Axiom for posets of cardinality (20)+ implies Projective Determinacy.
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