Mcgee on Open-ended Schemas

Let us start by rehearsing some standard definitions. Define the notion of a theory as follows: Definition 1 (Theory) Let A be the axioms of a theory T. The theory of T is the closure of A under semantic consequence, i.e. T = {φ : A φ}. For example, if T is Peano arithmetic, the theory of T is the axioms of Peano arithmetic together with all their semantic consequences. (Obviously , the axioms A of a theory T will always be included in the theory since φ φ for any φ.) 1 Define the notion of categoricity as follows: Definition 2 (Categoricity) A theory T is categorical if, and only if, any two models of T are isomorphic. The notions of a model and isomorphism relied on in Definition 2 are completely standard. An interpretation of a theory T is a structure I = d,i, where d is a non-empty set (the domain of T) and i is an interpretation function that assigns referents to the items of the language of T. A model of T is an interpretation which makes the axioms of T true (and so, all 1 Due to soundness and (deductive) completeness (cf. footnote 5), it will not matter in a first-order setting whether we speak in terms of semantic or syntactic, or deductive, consequence.