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Proceedings of the American Mathematical Society

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A large $ \Pi\sp 1\sb 2$ set, absolute for set forcings


Author: Sy D. Friedman
Journal: Proc. Amer. Math. Soc. 122 (1994), 253-256
MSC: Primary 03E15; Secondary 03E35
DOI: https://doi.org/10.1090/S0002-9939-1994-1231297-3
MathSciNet review: 1231297
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Abstract: We show how to obtain, by class-forcing over L, a set of reals X which is large in $ L(X)$ and has a $ \Pi _2^1$ definition valid in all set-generic extensions of $ L(X)$. As a consequence we show that it is consistent for the Perfect Set Property to hold for $ \Sigma _2^1$ sets yet fail for some $ \Pi _2^1$ set. Also it is consistent for the perfect set property to hold for $ \Sigma _2^1$ sets and for there to be a long $ \Pi _2^1$ well-ordering. These applications (necessarily) assume the consistency of an inaccessible cardinal.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1231297-3
Article copyright: © Copyright 1994 American Mathematical Society