✦ LIBER ✦

Complexity of reals in inner models of set theory

✍ Scribed by Boban Velickovic; W.Hugh Woodin

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 931 KB
Volume: 92
Category: Article
ISSN: 0168-0072
DOI: 10.1016/s0168-0072(98)00010-4

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✦ Synopsis

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 l-2, M 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 F, set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly Ni, 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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