Complexity of reals in inner models of set theory

Annals of Pure and Applied Logic 92 (3):283-295 (1998)
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Abstract

We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry

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Author's Profile

W. Hugh Woodin
Harvard University

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References found in this work

Internal cohen extensions.D. A. Martin & R. M. Solovay - 1970 - Annals of Mathematical Logic 2 (2):143-178.
One hundred and two problems in mathematical logic.Harvey Friedman - 1975 - Journal of Symbolic Logic 40 (2):113-129.
Internal Cohen extensions.D. A. Martin - 1970 - Annals of Mathematical Logic 2 (2):143.
Nonsplitting subset of κ.Moti Gitik - 1985 - Journal of Symbolic Logic 50 (4):881-894.
Forcing the failure of ch by adding a real.Saharon Shelah & Hugh Woodin - 1984 - Journal of Symbolic Logic 49 (4):1185-1189.

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