Cardinal-preserving extensions

Abstract

A classic result of Baumgartner-Harrington-Kleinberg implies that assuming CH a stationary subset of ω₁ has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω₁. Therefore, assuming that ω₂^L is countable: { X ∈ L | X⊆ ω₁^L and X has a CUB subset in a cardinal-preserving extension of L} is constructible, as it equals the set of constructible subsets of ω₁^L which in L are stationary. Is there a similar such result for subsets of ω₂^L? Building on work of M. Stanley, we show that there is not. We shall also consider a number of related problems, examining the extent to which they are “solvable” in the above sense, as well as defining a notion of reduction between them.