Forcing closed unbounded subsets of ω2 Original Research Article

It is shown that there is no satisfactory first-order characterization of those subsets of ω2 that have closed unbounded subsets in ω1,ω2 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.