  In 1904 Oswald Veblen described a theory as categorical if it has just one model up to isomorphism, i.e. it has a model and all its models are isomorphic to each other.  (The name was suggested to him by John Dewey, who also suggested the name ‘disjunctive’ for other theories.  This pair of terms come from traditional logic as names of types of sentence.) The depressing news is that there are no categorical first-order theories with infinite models; we can see this at once from the upward Löwenheim-Skolem theorem. In fact if T is a first-order theory with infinite models, then the strongest kind of categoricity we can hope for in T is that for certain infinite cardinals κ, T has exactly one model of cardinality κ, up to isomorphism. This property of T is called κ-categoricity. 