What Is a Number?

Abstract

In Chap. 1, we used addition and multiplication of the natural numbers to introduce first-order logic. Now, equipped with formal logic, we will go back and we will reconstruct the natural numbers and other number systems that are built on them. This looks circular, and to some extent it is. The set of natural numbers with a set of two relations—addition and multiplication—is a fundamental mathematical structure. In the previous discussion, we took the structure of natural numbers for granted, and we saw how some of its features can be described using first-order logic. Now we will examine the notion of natural number more carefully. It will not be as easy as one could expect.

In a scientific technique there is almost always an arbitrary element, and a philosophical discussion which puts too much stress on the ‘technical’ aspects of the problem in question, exposes itself all too easily to the suspicion of resting for a part on purely arbitrary stipulations. Evert W. Beth, Aspects of Modern Logic [ 5 ].