\(0^{\sharp}\) and elementary end extensions of \(V_{\kappa}\) (Q2719021)

A theorem of \textit{H. J. Keisler} and \textit{J. H. Silver} [Axiomatic set theory, Proc. Symp. Pure Math. 13, Part I, 177-187 (1971; Zbl 0225.02039)]\ says that if \(\kappa\) is weakly compact, then \((V_\kappa,\in)\) has a well-founded elementary end-extension. The author shows here that, however, when passing to a certain inner model \(N\) of \(L[0^\#]\), the model \((V^N_\kappa,\in)\) may fail to have any elementary end-extension. One way to do this, as demonstrated by the author, is to generically add a Souslin tree and a coding to the tree order over \(L\) so that a generic object could be produced in \(L[0^\#]\). In the minimal model \(N\) constructed from such generic object, such a tree witnesses that there is no elementary end extension of \(V^N_\kappa\).