On the categoricity of complete second order theories

TL;DR

This paper explores the conditions under which complete second order theories are categorical, showing that set-theoretic assumptions like PD influence categoricity, and that forcing can produce non-categorical theories even with large cardinals.

Contribution

It demonstrates how set-theoretic assumptions and forcing techniques affect the categoricity of complete second order theories, extending previous results by Solovay.

Findings

Assuming PD, all finitely axiomatized second order theories with a countable model are categorical.

Under PD, there exist non-categorical complete recursively axiomatized second order theories.

Forcing can produce non-categorical theories and influence categoricity at certain singular cardinals.

Abstract

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.