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 $n\ge 2$ . Then: 1. If it holds in the constructible universe $\mathbf{L}$ that $a\subseteq \omega$ and $a\notin {\Sigma }_n^1\cup {\Pi }_n^1$ , then there is a generic extension of $\mathbf{L}$ in which $a\in {\Delta }_{n+1}^1$ but still $a\notin {\Sigma }_n^1\cup {\Pi }_n^1$ , and moreover, any set $x\subseteq \omega$ , $x\in {\Sigma }_n^1$ , is constructible and ${\Sigma }_n^1$ in $\mathbf{L}$ . 2. There exists a generic extension $\mathbf{L}$ in which it is true that there is a nonconstructible ${\Delta }_{n+1}^1$ set $a\subseteq \omega$ , but all ${\Sigma }_n^1$ sets $x\subseteq \omega$ are constructible and even ${\Sigma }_n^1$ in $\mathbf{L}$ , and in addition, $\mathbf{V}=\mathbf{L}\left[a\right]$ in the extension. 3. There exists an generic extension of $\mathbf{L}$ in which there is a nonconstructible ${\Sigma }_{n+1}^1$ set $a\subseteq \omega$ , but all ${\Delta }_{n+1}^1$ sets $x\subseteq \omega$ are constructible and ${\Delta }_{n+1}^1$ in $\mathbf{L}$ . Thus, nonconstructible reals (here subsets of $\omega$ ) can first appear at a given lightface projective class strictly higher than ${\Sigma }_2^1$ , in an appropriate generic extension of $\mathbf{L}$ . The lower limit ${\Sigma }_2^1$ is motivated by the Shoenfield absoluteness theorem, which implies that all ${\Sigma }_2^1$ sets $a\subseteq \omega$ are constructible. Our methods are based on almost-disjoint forcing. We add a sufficient number of generic reals to $\mathbf{L}$ , which are very similar at a given projective level n but discernible at the next level $n+1$ .