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Cardinal-preserving extensions

Published online by Cambridge University Press:  12 March 2014

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 ω 1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω 1. Therefore, assuming that ω 2 L is countable: {XLXω 1 L and X has a CUB subset in a cardinal-preserving extension of L} is constructive, as it equals the set of constructible subsets of ω 1 L which in L are stationary. Is there a similar such result for subsets of ω 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 denning a notion of reduction between them.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2003

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References

[1] Baumgartner, J., Harrington, L., and Kleinberg, E., Adding a closed unbounded set, this Journal, vol. 41 (1976), pp. 481482.Google Scholar
[2] Devlin, K. and Jensen, R., Marginalia to a theorem of Silver, Lecture Notes in Mathematics, vol. 499, Springer-Verlag, 1975.Google Scholar
[3] Friedman, S., The Π2 1-singleton conjecture, Journal of the American Mathematical Society, vol. 3 (1990), pp. 771791.Google Scholar
[4] Friedman, S., Generic saturation, this Journal, vol. 63 (1998), pp. 158162.Google Scholar
[5] Friedman, S., Fine structure and class forcing, Series in Logic and its Applications, de Gruyter, 2000.CrossRefGoogle Scholar
[6] Harrington, L. and Kechris, A., Π2 1-singletons and 0# , Fundamenta Mathematicae, vol. 95 (1977), no. 3, pp. 167171.CrossRefGoogle Scholar
[7] Jensen, R., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.CrossRefGoogle Scholar
[8] Martin, D. A. and Solovay, R. M., A basis theorem for Σ3 1 sets of reals, Annals of Mathematics, vol. 89 (1969), no. 2, pp. 138160.CrossRefGoogle Scholar
[9] Stanley, M., Forcing closed unbounded subsets of ω 2, to appear.Google Scholar
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