Modulo Infinity

Abstract

The classical consistent complete denumerable model of the natural numbers, {0,1, 2,...}, of order type ω, is also called the standard model of classical Peano arithmetic. This contrasts with the (classical consistent complete denumerable) nonstandard model. As is well known the latter has a domain of order type ω + η(ω* + ω), consisting of an initial block isomorphic to {0,1, 2,...} (called the finite natural numbers), with succeeding blocks of numbers (called the infinite natural numbers) isomorphic to the integers (order type ω*+ω), the blocks themselves being densely ordered (order type η). Both models verify exactly the sentences of classical standard arithmetic P## (see Definition 2.3) in their common language. In this chapter we consider consistent and inconsistent theories which arise from the nonstandard model.