The Real Numbers

Abstract

To review our progress to date, we have rigorously constructed three fundamental number systems: the natural number system ${{\textbf{N}}}$, the integers ${{\textbf{Z}}}$, and the rationals ${{\textbf{Q}}}$. We defined the natural numbers using the five Peano axioms and postulated that such a number system existed; this is plausible, since the natural numbers correspond to the very intuitive and fundamental notion of sequential counting. Using that number system one could then recursively define addition and multiplication and verify that they obeyed the usual laws of algebra. We then constructed the integers by taking formal differences of the natural numbers, $a {\,\textemdash \,}b$. We then constructed the rationals by taking formal quotients of the integers, a//b, although we need to exclude division by zero in order to keep the laws of algebra reasonable. (You are of course free to design your own number system, possibly including one where division by zero is permitted; but you will have to give up one or more of the field axioms from Proposition 4.2.4, among other things, and you will probably get a less useful number system in which to do any real-world problems.).