For each σ-structure A, there are several associated theories in a larger signature σ' that extends σ by adding one new constant symbol for each element of the domain of A. (If the new constant symbols are identified with the elements of A that they represent, σ' can be taken to be σ \cup A.) The cardinality of σ' is thus the larger of the cardinality of σ and the cardinality of A.
