standard model a term that, like ‘non-standard model’, is used with regard to theories that systematize (part of) our knowledge of some mathematical structure, for instance the structure of natural numbers with addition, multiplication, and the successor function, or the structure of real numbers with ordering, addition, and multiplication. Models isomorphic to this intended mathematical structure are the ‘standard models’ of the theory, while any other, non-isomorphic, model of the theory is a ‘non-standard’ model. Since Peano arithmetic is incomplete, it has consistent extensions that have no standard model. But there are also non-standard, countable models of complete number theory, the set of all true first-order sentences about natural numbers, as was first shown by Skolem in 1934. Categorical theories do not have a non-standard model. It is less clear whether there is a standard model of set theory, although a countable model would certainly count as non-standard. The Skolem paradox is that any first-order formulation of set theory, like ZF, due to Zermelo and Fraenkel, has a countable model, while it seems to assert the existence of non-countable sets. Many other important mathematical structures cannot be characterized by a categorical set of first-order axioms, and thus allow non-standard models. The American philosopher Putnam has argued that this fact has important implications for the debate about realism in the philosophy of language. If axioms cannot capture the ‘intuitive’ notion of a set, what could? Some of his detractors have pointed out that within second-order logic categorical characterizations are often possible. But Putnam has objected that the intended interpretation of second-order logic itself is not fixed by the use of the formalism of second-order logic, where ‘use’ is determined by the rules of inference for second-order logic we know about. Moreover, categorical theories are sometimes uninformative.
See also CATEGORICAL THEORY, GöDEL’s INCOMPLETENESS THEOREMS , SET THEOR. Z.G.S.