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

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 .

AN ANALYSIS OF THE MODELS

We analyze the models $L[T_{2n} ]$, where $T_{2n}$ is a tree on $\omega \times \kappa _{2n + 1}^1 $ projecting to a universal ${\rm{\Pi }}_{2n}^1 $ set of reals, for $n > 1$. Following Hjorth’s work on $L[T_2 ]$, we show that under ${\rm{Det}}\left( {{\rm{}}_{2n}^1 } \right)$, the models $L[T_{2n} ]$ are unique, that is they do not depend of the choice of the tree $T_{2n}$. This requires a generalization of the Kechris–Martin theorem to all pointclasses${\rm{\Pi }}_{2n + 1}^1$. We then characterize these models as constructible models relative to the direct limit of all countable nondropping iterates of${\cal M}_{2n + 1}^\# $. We then show that the GCH holds in $L[T_{2n} ]$, for every $n < \omega $, even though they are not extender models. This analysis localizes the HOD analysis of Steel and Woodin at the even levels of the projective hierarchy.

Universal sets for pointsets properly on the nth level of the projective hierarchy

The Axiom of Projective Determinacy implies the existence of a universal set for every n ≥ 1. Assuming there exists a universal set. In ZFC there is a universal set for every α.

Expansions of the real field by open sets: definability versus interpretability

An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in (, E) either has interior or is Hausdorff null.