We show that the tree property, stationary reflection and the failure of approachability at \(\kappa ^{++}\) are consistent with \(\mathfrak {u}(\kappa )= \kappa ^+ < 2^\kappa \), where \(\kappa \) is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if \(\lambda \) is a regular cardinal, then stationary reflection at \(\lambda ^+\) is indestructible under all \(\lambda \)-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).

中文翻译：

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小的$$ \ mathfrak {u}（\ kappa）$$ u（κ）为奇数$$ \ kappa $$κ，紧凑度为$$ \ kappa ^ {++} $$κ+ +

我们显示树的性质，固定反射和\（\ kappa ^ {++} \）处的可接近性失败与\（\ mathfrak {u}（\ kappa）= \ kappa ^ + <2 ^ \ kappa一致\），其中\（\ kappa \）是具有可数或不可数同素性的奇异强极限基数。作为副产品，我们表明如果\（\ lambda \）是常规基数，则在所有\（\ lambda \）- cc强迫下（在一般情况下）在\（\ lambda ^ + \）处的静态反射是不可破坏的。有趣的是，我们还陈述了保持球杆静止反射的相关结果。