The Fraenkel‐Carnap question for Dedekind algebras

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.