It is shown that there is no satisfactory first-order characterization of those subsets of ω 2 that have closed unbounded subsets in ω 1 , ω 2 and GCH preserving outer models. These “anticharacterization” results generalize to subsets of successors of uncountable regular cardinals. Similar results are proved for trees of height and cardinality κ + and for partitions of [ κ + ] 2 , when κ is an infinite cardinal.

Stanley, M.C., A Π12 singleton incompatible with 0#, Annals of Pure and Applied Logic 66 27–88. A non-constructible Π12 singleton that is absolute for ω-models of ZF is produced by class forcing over the minimum model.