Abstract
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 .
Highlights
The lower limit Σ12 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ12 sets a ⊆ ω are constructible
Problems of definability and effective construction of mathematical objects have always been in the focus of attention during the development of mathematical foundations
The general goal of the research line of this paper is to explore the existence of effectively definable structures in descriptive set theory on specific levels of the projective hierarchy
Summary
Σ1n+1 set a ⊆ ω , but all ∆1n+1 sets x ⊆ ω are constructible and ∆1n+1 in L. Nonconstructible reals (here subsets of ω ) can first appear at a given lightface projective class strictly higher than. Σ12 , in an appropriate generic extension of L. The lower limit Σ12 is motivated by the Shoenfield absoluteness theorem, which implies that all Σ12 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 level n + 1
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