It is known that if κ is inaccessible in V , and W is an outer model of V such that (κ +) V = (κ +) W , and cf W (κ) = ω, then κ,ω holds in W. Many strengthenings of this theorem have been investigated as well. We show that this theorem does not generalize to uncountable cofinalities: There is a model V in which κ is inaccessible and there is a forcing extension W of V in which (κ +) V = (κ +) W , ω < cf W (κ) < κ, and κ,τ fails in W for all τ < κ. We make use of Magidor's forcing for singularizing an inaccessible κ to have uncountable cofinality. Along the way, we analyze stationary reflection in this model, and we show that it is possible for κ,cf(κ) to hold in a forcing extension by Magidor's poset if the ground model is prepared with a partial square sequence.

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals κ, updating some classical constructions in the process. This includes models of CSR(κ ++ ) ∧ TP(κ ++ ) (both with and without AP(κ ++ )) and models of the conjunctions SR(κ ++ ) ∧ wTP(κ ++ ) ∧ AP(κ ++ ) and ¬AP(κ ++ ) ∧ SR(κ ++ ) (the latter was originally obtained in joint work by Krueger and the first author [8], and is here given using different methods). Analogs of these results with the failure of SH(κ ++ ) are given as well. Finally, we obtain all of our results with an arbitrarily large 2 κ , applying recent joint work by Honzik and the third author.

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Current research in set theory raises the possibility that κ,<λ can be made compatible with some stationary reflection, depending on the parameter λ. The purpose of this paper is to demonstrate the difficulty in such results. We prove that the poset S(κ, < λ), which adds a κ,<λ -sequence by initial segments, will also add non-reflecting stationary sets concentrating in any given cofinality below κ. We also investigate the CMB poset, which adds * κ in a slightly different way. We prove that the CMB poset also adds non-reflecting stationary sets, but not necessarily concentrating in any cofinality.

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