Journal of Symbolic Logic 68 (4):1163-1170 (2003)

A 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 $\omega_2^L$ is countable: { $X \in L \mid X \subseteq \omega_1^L$ and X has a CUB subset in a cardinal -preserving extension of L} is constructible, as it equals the set of constructible subsets of $\omega_1^L$ which in L are stationary. Is there a similar such result for subsets of $\ omega_2^L$ ? 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 defining a notion of reduction between them
Keywords Descriptive set theory   large cardinals   innermodels
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DOI 10.2178/jsl/1067620178
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References found in this work BETA

The Fine Structure of the Constructible Hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.

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Citations of this work BETA

Adding Closed Unbounded Subsets of Ω₂ with Finite Forcing.William J. Mitchell - 2005 - Notre Dame Journal of Formal Logic 46 (3):357-371.
Forcing Closed Unbounded Subsets of Ω2.M. C. Stanley - 2001 - Annals of Pure and Applied Logic 110 (1-3):23-87.

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