A non-generic real incompatible with \(0^\sharp\) (Q1361253)

Let \(L\) be a minimal countable standard transitive model of \(\text{ZFC}+{V=L}\) which means that \(L=L_\beta\), \(L\) is a model of ZFC, and for some \(\alpha<\beta\) the elements of \(L_\beta\) are first-order definable from ordinals less than \(\alpha\). A real \(x_{\text{ng}}\in{}^\omega2\) is constructed such that \(x_{\text{ng}}\notin L\), \(L[x_{\text{ng}}]\) satisfies ZFC, and in a certain sense \(x_{\text{ng}}\) is not generic over any outer model of \(L\) containing the same ordinals as \(L\). By another result of the author [``Class forcing'', to appear] it follows immediately that there is no outer model \(V\), a \(V\)-amenable partial ordering \(\mathbb P\) and a filter \(G\) on \(\mathbb P\) meeting every \(\langle V;{\mathbb P}\rangle\)-definable dense subclass of \(\mathbb P\) such that \(x_{\text{ng}}\in V[G]\setminus V\) and \(\langle V[G];V,P,G\rangle\) satisfies ZFC. For the proof some trees \(T^n_{k,i}\) of closed sets of ordinals (uniformly in \(k\) and \(i\) but not in \(n\)) for all \(n,k\in\omega\) and \(i=0\), 1, class orderings \({\mathbb P}^n\) for \(n\in\omega\) with \({\mathbb P}^{n+1}\subseteq{\mathbb P}^n\), and a Shoenfield term \({\overset\circ x}\) common to the forcing languages of all orderings \({\mathbb P}^n\) are defined in \(L\) so that if \(G\) is sufficiently \({\mathbb P}^n\)-generic, then \(x_{\text{ng}}={\overset\circ x}^G\) codes a suitably generic branch through \(T^n_{k,i}\) where \({\overset\circ x}^G(k)=i\). The coding is based on a simplified version of Jensen coding of generic extension of \(L\). The motivation for the main result was a Beller-Jensen-Welch conjecture saying that if \(x\) is a real and \(0^\sharp\in L[x]\), then \(x\) is class generic over \(L\). The construction of the real \(x_{\text{ng}}\) is incompatible with the existence of \(0^\sharp\) (which implies that \(L\) is not minimal). The author formulates a list of open questions on the possibility of constructions of reals from \(0^\sharp\) not being generic in certain sense.