Definability of initial segments (Q1430996)

Let \(L\) be a first-order countable language, \(M\) an \(L\)-structure and \(P\) a unary second-order variable. A subset \(A\) of \(M\) is called implicitly definable in \(M\) if there exists an \((L \cup \{P\})\)-sentence \(\phi(P)\) with parameters such that \(A\) is the unique solution to \(\phi(P)\), i.e., \(\{A\}=\{B \subseteq M : M \models \phi(B)\}\). The main result states that there is a model of the Peano arithmetic in which the standard part is implicitly definable.