Abstract
Suppose that T⁎ is an ω1-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA(T⁎) for proper forcings which preserve these properties of T⁎. We prove that PFA(T⁎) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ω1, and the P-ideal dichotomy. On the other hand, PFA(T⁎) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.
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