Ordinal definable subsets of singular cardinals

A remarkable result by Shelah states that if κ is a singular strong limit cardinal of uncountable cofinality, then there is a subset x of κ such that HODx contains the power set of κ. We develop a version of diagonal extender-based supercompact Prikry forcing, and use it to show that singular cardinals of countable cofinality do not in general have this property, and in fact it is consistent that for some singular strong limit cardinal κ of countable cofinality κ+ is supercompact in HODx for all x ⊆ κ.

谢拉（Shelah）的一个显著结果表明，如果κ是具有不可数共尾性的奇异强极限基数，那么存在κ的一个子集x，使得HODx包含κ的幂集。我们发展了一种基于对角扩张子的超紧普利克里（Prikry）力迫的变体，并利用它表明具有可数共尾性的奇异基数通常不具有这种性质，并且实际上存在这样的一致性：对于某些具有可数共尾性的奇异强极限基数κ，对于所有x⊆κ，κ⁺在HODx中是超紧的。