Discretely ordered rings
Richard W. Kaye · 1991
Abstract We start our study of models of arithmetic in earnest by writing down some simple axioms that are obviously true in ℕ. The resulting theory, PA− , is described in Section 2.1 and is our ‘base theory’ to which we shall later add induction axioms. PA- has some very nice features which we explore in this chapter. In particular we will show in Section 2.2 that any model of PA- may be regarded as an end-extension of the standard model, and we introduce the notion of a Σ1 sentence, and show that PA- is strong enough to prove all true Σ1 sentences.