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Cardinal-preserving extensions

Published online by Cambridge University Press:  12 March 2014

Sy D. Friedman
Sy D. Friedman*
Affiliation:
Institute for Logic, University of Vienna, Waehringer Strasse 25, A-1090 Vienna, Austria Mathematics Department, MIT, Cambridge, Massachusetts 02139, USA, E-mail: sdf@logic.univie.ac.at
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Abstract

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 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.

Keywords

Descriptive set theorylarge cardinalsinnermodels
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1163 - 1170
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

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