The Fraenkel-carnap Question for Limited Higher-order Languages

In the late 20’s Fraenkel and Carnap raised the question of whether or not all semantically complete finitely axiomatizable theories formulated in the theory of types are categorical. Carnap announced a positive answer to this question. However, that proof was flawed (see Awody and Carus [2001] and Awodey and Reck [2002]). The Fraenkel-Carnap question has been ignored until it was rediscovered by Awodey, Carus and Reck. Scott noticed that the question has a positive answer for theories formulated in those n-order languages (n ≥ 2) whose non-logical vocabulary is empty. It is shown here that this question also has a positive answer when the non-logical vocabulary is non-empty provided it includes only individual constants or monadic predicate constants.