 There is a strong form of categoricity which holds for Henkin structures in important cases and agrees with the usual concept of categoricity in the case of full Henkin models. It builds on the remarkable ability of second-order logic to express its own categoricity. The isomorphism \((M,R)\cong(M',R')\) of two structures \((M,R)\) and \((M',R')\), where the predicates R and \(R'\) are, say, binary, can be expressed in second-order logic as follows, letting \(\ma=(A,M,M',R,R')\), where A is any set containing \(M\cup M'\), \(M=U^\ma\), \(M'={U'}^\ma\), \(R=P^\ma\) and \(R'=P'^\ma\). Now 