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Σ1(κ)-DEFINABLE SUBSETS OF H(κ +)

  • PHILIPP LÜCKE (a1), RALF SCHINDLER (a2) and PHILIPP SCHLICHT (a3)

Abstract

We study Σ1(ω 1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω 1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω 1)-definable, the set of all stationary subsets of ω 1 is not Σ1(ω 1)-definable and the complement of every Σ1(ω 1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω 1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω 1)-definable well-ordering of H(ω 2) and the existence of a Δ1(ω 1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ . We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω 1)-definable uniformization of the club filter on ω 1. Moreover, we prove a perfect set theorem for Σ1(ω 1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$ , assuming that there is a measurable cardinal and the nonstationary ideal on ω 1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κ κ, in the case where κ itself has certain large cardinal properties.

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