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Projectively well-ordered inner models. (English) Zbl 0821.03023
Summary: We show that the reals in the minimal iterable inner model having \(n\) Woodin cardinals are precisely those which are \(\Delta^ 1_{n+2}\) definable from some countable ordinal. (One direction here is due to Hugh Woodin.) It follows that this model satisfies “There is a \(\Delta^ 1_{n+2}\) well-order of the reals”. We also describe some other connections between the descriptive set theory of projective sets and inner models with finitely many Woodin cardinals.

MSC:
03E15 Descriptive set theory
03E60 Determinacy principles
03E55 Large cardinals
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