Models of Second-order Arithmetic
Recall that we view second-order arithmetic as a theory in first-order predicate calculus. Thus a model of the language of second-order arithmetic consists of a set M (which forms the range of individual variables) together with a constant 0 (an element of M), a function S from M to M, two binary operations + and · on M, a binary relation < on M, and a collection D of subsets of M, which is the range of the set variables. By omitting D we obtain a model of the language of first order arithmetic.
When D is the full powerset of M, the model is called a full model. The use of full second-order semantics is equivalent to limiting the models of second-order arithmetic to the full models. In fact, the axioms of second-order arithmetic have only one full model. This follows from the fact that the axioms of Peano arithmetic with the second-order induction axiom have only one model under second-order semantics.
When M is the usual set of natural numbers with its usual operations, is called an ω-model. In this case we may identify the model with D, its collection of sets of naturals, because this set is enough to completely determine an ω-model.
The unique full model, which is the usual set of natural numbers with its usual structure and all its subsets, is called the intended or standard model of second-order arithmetic.
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