which consists of P along with a unary function symbol f. Let be the class of all -structures such that f is a bijection and P holds for x iff P does not hold for f(x). is clearly an elementary class, and therefore K is an example of a pseudo-elementary class that is not elementary. Non-pseudo-elementary class Let σ be an arbitrary signature. The class K of all finite σ-structures is not elementary, because (as shown above) its complement is elementary but not basic elementary. Since this is also true for every signature extending σ, K is not even a pseudo-elementary class. This example demonstrates the limits of expressive power inherent in first-order logic as opposed to the far more expressive second-order logic. Second-order logic, however, fails to retain many desirable properties of first-order logic, such as the completeness and compactness theorems. References Model theory