A version of the Jensen-Johnsbråten coding at arbitrary level $n\ge 3$

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 ${\Delta }_{n+1}^1$ in $L[a]$; and 2) if $b\in L[a],$ $b\subseteq \omega$ is ${\Sigma }_n^1$ in $L[a]$ then $b\in L$ and $b$ is ${\Sigma }_n^1$ in $L$. In addition, if a model $M$ extends $L$ and contains two different $P$-generic sets $a,a^{\prime }\subseteq \omega ,$ then ${\omega }_1^M>{\omega }_1^L$. Comment: For $n=2,$ this is a result of Jensen and Johnsbråten, 1974. In this case, 2) is a corollary of the Shoenfield absoluteness theorem.