Club stationary reflection and the special Aronszajn tree property

Abstract We prove that it is consistent Club Stationary Reflection and the Special Aronszajn Tree Property simultaneously hold on $\omega _2$ , thereby contributing to study of tension between compactness incompactness in set theory. The poset which produces final model follows collapse an ineffable cardinal first with iteration club adding (with anticipation) second specializing trees. In part paper, we a general theorem about trees after forcing what call $\mathcal {F}$ -Strongly Proper posets, where either weakly compact filter or dual ineffability ideal. This type poset, Levy degenerate example, uses systems exact residue functions create many strongly generic conditions. new result stationary preservation by quotients this kind poset; as corollary, show original Laver–Shelah model, starts from cardinal, satisfies strong reflection principle, although fails satisfy full Reflection. part, composition collapsing poset. After proving tree preservation, how obtain model.