Small u(kappa) at singular kappa with compactness at kappa++

Abstract

We show that the tree property, stationary reflection and the failure of approachability at κ ++ are consistent with u(κ) = κ + < 2 κ , where κ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if λ is a regular cardinal, then stationary reflection at λ + is indestructible under all λ-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection). Contents1. Introduction 1 2. Preliminaries 3 2.1. Compactness properties 3 2.2. Branch lemmas 4 2.3. Some pcf notions 5 2.4. Prikry and Magidor forcing 5 3. A review of the arguments for a small u(κ) 6 4. Countable cofinality 9 4.1. Definition of forcing 9 4.2. Small ultrafilter number 11 4.3. Tree property 11 4.4. Stationary reflection 14 4.5. Failure of approachability 17 5. Uncountable cofinalities 17 6. Open questions 18 References 19