Noncharacterizability of the Syntax Set

Introduction. Craig showed in [3] that Beth's theorem fails for 2nd order logic. This is so since the syntax for 2nd order logic can be represented in the structure X of natural numbers and X itself can be characterized in 2nd order logic. Thus the satisfaction relation S is implicitly definable yet not explicitly definable (by a Tarski diagonal argument) and Beth fails. Kreisel pointed out in a review of Craig's paper that a similar argument shows that Beth's theorem fails for cW-logic or for any logic whose syntax can be represented in X and in which X can be characterized. In this paper we use the abstract model-theoretic notions of truth adequacy, truthmaximality and truth-completeness developed by Feferman in [4] to prove the following generalization of Craig's result: If L is a truth-complete logic, then no structure in which the syntax is represented is L-characterizable. This is applied via a corollary to give new examples of failures of Beth's theorem. Another way to construe this result is that truth-complete logics L have the desirable modeltheoretic property (see [5]) that the structure in which the syntax is represented is not generalized finite (interpreting generalized finite as L-characterizability).