## Numbers

The basic set theory allows us to construct a set with a function “next”, denoted by , which is bijective. This set describes the process of counting objects and is the most basic structure. Starting from a distinguished element denoted by 1, we construct an infinite number of elements etc. There are two axioms guaranteeing that the set indeed coincides with what we call the set of natural numbers:

- Any element is obtained by the iteration of : .

Using this function and its partial inverse one can define on the order and the operations of addition (as repeated addition of 1 which is just evaluation of ) and multiplication (repeated addition).

However, not all equations of the form or are solvable. One can enlarge by adding solutions of all such equations, obtaining the set of integer numbers which is a commutative group with respect to the operation of addition, and finally the set of rational numbers in which division is available by any nonzero number.

Division by zero is impossible: if we add “solution of the equation ” as a new imaginary element, then we will not be able to do some arithmetic operations on it. Still, if we are ready to pay this price, then the rational numbers can be extended by a new element so that, say, the function would be everywhere defined and continuous.

Details are available in the lecture notes here.