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
ω2L is countable: { X ∈ L |
X⊆ ω1L and
X has a CUB subset in a cardinal-preserving extension of
L} is
constructible, as it equals the set of constructible subsets of
ω1L which in L are
stationary. Is there a similar such
result for subsets of ω2L? 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.