On the categoricity of complete second order theories

We show, assuming PD, that every complete finitely axiomatized second order theory with a countable model is categorical, but that there is, assuming again PD, a complete recursively axiomatized second order theory with a countable model which is non-categorical. We show that the existence of even very large (e.g. supercompact) cardinals does not imply the categoricity of all finite complete second order theories. More exactly, we show that a non-categorical complete finitely axiomatized second order theory can always be obtained by (set) forcing. We also show that the categoricity of all finite complete second order theories with a model of a certain singular cardinality kappa of uncountable cofinality can be forced over any model of set theory. Previously, Solovay had proved, assuming V=L, that every complete finitely axiomatized second order theory (with or without a countable model) is categorical, and that in a generic extension of L there is a complete finitely axiomatized second order theory with a countable model which is non-categorical.