A version of the Jensen-Johnsbr{\aa}ten coding at arbitrary level $n\geq 3$

Abstract

Theorem: Let $n\ge 2.$ There is a CCC in $L$ forcing notion $P=P_n\in L$ such that $P$-generic extensions of $L$ are of the form $L[a],$ where $a\subseteq\omega$ and 1) $a$ is $\Delta^1_{n+1}$ in $L[a]$; and 2) if $b\in L[a],$ $b\subseteq\omega$ is $\Sigma^1_n$ in $L[a]$ then $b\in L$ and $b$ is $\Sigma^1_n$ in $L$. In addition, if a model $M$ extends $L$ and contains two different $P$-generic sets $a,\,a'\subseteq\omega,$ then $\omega^M_1 > \omega^L_1$. Comment: For $n=2,$ this is a result of Jensen and Johnsbr{\aa}ten, 1974. In this case, 2) is a corollary of the Shoenfield absoluteness theorem.