Subsets of ℵ_1 constructible from a real

Abstract

The purpose of this paper is to give a necessary and sufficient condition for a subset of ℵ_1 to be constructible from a real in terms of structural properties of the code set of A, valid under the condition that an appropriate measurable cardinal exists. This can be combined with recent results of Woodin to provide upper bounds for the consistency strength, of theories of the form ZFC + ∀x Є ω^ω(x^# exists)+ "every subset of ℵ_1 with code set in Г is constructible from a real," for various pointclasses Г.

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