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Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level

Institute for Information Transmission Problems of the Russian Academy of Sciences, 127051 Moscow, Russia
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These authors contributed equally to this work.
Mathematics 2020, 8(6), 910; https://doi.org/10.3390/math8060910
Received: 11 May 2020 / Revised: 25 May 2020 / Accepted: 27 May 2020 / Published: 3 June 2020
(This article belongs to the Special Issue Mathematical Logic and Its Applications 2020)
Models of set theory are defined, in which nonconstructible reals first appear on a given level of the projective hierarchy. Our main results are as follows. Suppose that n 2 . Then: 1. If it holds in the constructible universe L that a ω and a Σ n 1 Π n 1 , then there is a generic extension of L in which a Δ n + 1 1 but still a Σ n 1 Π n 1 , and moreover, any set x ω , x Σ n 1 , is constructible and Σ n 1 in L . 2. There exists a generic extension L in which it is true that there is a nonconstructible Δ n + 1 1 set a ω , but all Σ n 1 sets x ω are constructible and even Σ n 1 in L , and in addition, V = L [ a ] in the extension. 3. There exists an generic extension of L in which there is a nonconstructible Σ n + 1 1 set a ω , but all Δ n + 1 1 sets x ω are constructible and Δ n + 1 1 in L . Thus, nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than Σ 2 1 , in an appropriate generic extension of L . The lower limit Σ 2 1 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ 2 1 sets a ω are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to L , which are very similar at a given projective level n but discernible at the next level n + 1 . View Full-Text
Keywords: definability; nonconstructible reals; projective hierarchy; generic models; almost disjoint forcing definability; nonconstructible reals; projective hierarchy; generic models; almost disjoint forcing
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Kanovei, V.; Lyubetsky, V. Models of Set Theory in which Nonconstructible Reals First Appear at a Given Projective Level. Mathematics 2020, 8, 910.

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