categorical theory a theory all of whose models are isomorphic. Because of its weak expressive power, in first-order logic with identity only theories with a finite model can be categorical; without identity no theories are categorical. A more interesting property, therefore, is being categorical in power: a theory is categorical in power a when the theory has, up to isomorphism, only one model with a domain of cardinality a. Categoricity in power shows the capacity to characterize a structure completely, only limited by cardinality. For example, the first-order theory of dense order without endpoints is categorical in power w the cardinality of the natural numbers. The first-order theory of simple discrete orderings with initial element, the ordering of the natural numbers, is not categorical in power w. There are countable discrete orders, not isomorphic to the natural numbers, that are elementary equivalent to it, i.e., have the same elementary, first-order theory. In first-order logic categorical theories are complete. This is not necessarily true for extensions of first-order logic for which no completeness theorem holds. In such a logic a set of axioms may be categorical without providing an informative characterization of the theory of its unique model. The term ‘elementary equivalence’ was introduced around 1936 by Tarski for the property of being indistinguishable by elementary means. According to Oswald Veblen, who first used the term ‘categorical’ in 1904, in a discussion of the foundations of geometry, that term was suggested to him by the American pragmatist John Dewey. See also COMPLETE – NESS , MODEL THEORY. Z.G.S. categoricity, the semantic property belonging to a set of sentences, a ‘postulate set,’ that implicitly defines (completely describes, or characterizes up to isomorphism) the structure of its intended interpretation or standard model. The best-known categorical set of sentences is the postulate set for number theory attributed to Peano, which completely characterizes the structure of an arithmetic progression. This structure is exemplified by the system of natural numbers with zero as distinguished element and successor (addition of one) as distinguished function. Other exemplifications of this structure are obtained by taking as distinguished element an arbitrary integer, taking as distinguished function the process of adding an arbitrary positive or negative integer and taking as universe of discourse (or domain) the result of repeated application of the distinguished function to the distinguished element. (See, e.g., Russell’s Introduction to the Mathematical Philosophy, 1918.) More precisely, a postulate set is defined to be categorical if every two of its models (satisfying interpretations or realizations) are isomorphic (to each other), where, of course, two interpretations are isomorphic if between their respective universes of discourse there exists a one-to-one correspondence by which the distinguished elements, functions, relations, etc., of the one are mapped exactly onto those of the other. The importance of the analytic geometry of Descartes involves the fact that the system of points of a geometrical line with the ‘left-of relation’ distinguished is isomorphic to the system of real numbers with the ‘less-than’ relation distinguished. Categoricity, the ideal limit of success for the axiomatic method considered as a method for characterizing subject matter rather than for reorganizing a science, is known to be impossible with respect to certain subject matters using certain formal languages. The concept of categoricity can be traced back at least as far as Dedekind; the word is due to Dewey. See also AXIOMATIC METHOD, LÖWENHEIM – SKOLEM THEOREM , MATHEMATICAL ANALY – SIS , MODEL THEORY. J.COR.