A functorial \(\pi ^ 1_ 2\) singleton
If \(\phi\) is a \(\Sigma_ 1\) ZF-formula, M a transitive model of \(V=L\) such that every cardinal satisfies \(\phi\), then there is a cardinal and cofinality preserving class forcing such that in the enlarged model, \(V=L[r]\) holds for some real r, \(0^{\#}\) does not hold, and if \(L_{\alpha}[r]\) is admissible, then \(\phi\) holds for \(\alpha\). The proof uses the technique of Jensen coding.
cardinal and cofinality preserving class forcing
scientific article; zbMATH DE number 4118360