BU352 Study Guide - Final Guide: Peano Axioms, Unary Function
Document Summary
De nition 15. 1 a axiom is a formula that is assumed as a premise in any proof. An axiom schema is a set of axioms de ned by a pattern or rule. Fix the domain as n, the natural numbers. Interpret the constant symbol 0 as zerp and the unary function symbol s as success. s(x) x + 1. Thus each number in n has a term : 0, s(0), s(s(0)), . Zero and successor satisfy the following axioms: pa1 : xs(x) 6= 0, zero is not a successor , pa2 : x y((s(x) = s(y) x = y). , nothing has two predecessors. Induction axiom : pa7 : for each formula and variable x. The peano axioms imply all of the familiar properties of the natural numbers. Theorem 15. 2 p a x y(x + y = y + x) Lemma 15. 3 y(x + y = y + x) p a y(s(x) + y = y + s(x))
