A forcing axiom for a non-special Aronszajn tree

Annals of Pure and Applied Logic 171 (8):102820 (2020)
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Abstract

Suppose that T^∗ is an ω_1-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 ω_1, 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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References found in this work

An variation for one souslin tree.Paul Larson - 1999 - Journal of Symbolic Logic 64 (1):81-98.
Applications of cohomology to set theory I: Hausdorff gaps.Daniel E. Talayco - 1995 - Annals of Pure and Applied Logic 71 (1):69-106.
Suslin's hypothesis does not imply stationary antichains.Chaz Schlindwein - 1993 - Annals of Pure and Applied Logic 64 (2):153-167.

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