  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.
