The Fraenkel-Carnap question for Dedekind algebras

摘要

It is shown that the second-order theory of a Dedekind algebra is categorical if it is finitely axiomatizable. This provides a partial answer to an old and neglected question of Fraenkel and Carnap: whether every finitely axiomatizable semantically complete second-order theory is categorical. It follows that the second-order theory of a Dedekind algebra is finitely axiomatizable iff the algebra is finitely characterizable. It is also shown that the second-order theory of a Dedekind algebra is quasi-finitely axiomatizable iff the algebra is quasi-finitely characterizable.