Classical Arithmetization

Abstract

This chapter describes how one proceeds from natural to rational numbers, then to real and complex numbers, and to continuous functions—thus arithmetizing the foundations of analysis and geometry. The definitions of integers and rational numbers show why questions about them can, in principle, be reduced to questions about natural numbers and their addition and multiplication. This is what it means to say that the natural numbers are a foundation for the integer and rational numbers. But the next steps in the arithmetization project go beyond algebra. By admitting sets of rational numbers, one can enlarge the number system to one that admits certain infinite operations, such as forming infinite sums. This is crucial to building a foundation for analysis. As such, the chapter turns to the foundations of the natural numbers themselves, the “Peano axioms,” which gives a first glimpse of the logic underlying the arithmetization project.