The categoricity of a sentence θ(P) — that any two models of θ(P) are isomorphic — can therefore be written as the second-order sentence categ(θ(P)).

Solovay has made a posting in FOM (2006 Other Internet Resources) in which he shows that the related statement that every complete second-order sentence \(\theta\) is categorical, is independent of ZFC. 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)\co