Formalized Number Theory

In the preface to Principia Mathematica, it is stated that “what were formerly taken, tacitly or explicitly, as axioms, are either unnecessary or demonstrable” .1 In this spirit, we shall in this chapter define the natural numbers as the finite cardinals, and derive Peano’s Postulates from an Axiom of Infinity. We then establish those results about natural numbers and recursive functions which we need in order to establish the incompleteness results in Chapter 7.