Cardinal-preserving extensions

Abstract

AbstractA classic result of Baumgartner-Harrington-Kleinberg [1] 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 constructive, 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 [9], 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 denning a notion of reduction between them.