Peano's Axioms

authored by Premmi and Beguène Previous Topic: An Axiomatic Study of Numbers Introduction Thinking of numbers intuitively brings to mind the simplest and most fundamental set of numbers, namely the set of natural numbers. These numbers are used to count objects like cars, books, pens, etc. If we associate natural numbers such as 1, 2, 3, etc. with counting, then with what corresponding concepts do we relate numbers like -4, \sqrt{3} \text{ and } \frac{22}{7}? To reason about all kinds of numbers encountered during our study of mathematics, we need a precise mathematical framework for defining numbers. We will build this framework by first rigorously defining natural numbers axiomatically without relying on the intuitive notion of counting. Then, using this framework as a foundation, we will construct all the other sets of numbers such as integers, rational numbers, real numbers and complex numbers in terms of the natural numbers. A good axiomatic system assumes as little as possib ... Read full article.