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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:
 3e+15 Descriptive set theory 3e+60 Determinacy principles 3e+55 Large cardinals
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##### References:
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