Peano Axioms

is a natural number.

Each natural number has a successor .

Two natural numbers are equal if their successors are equal.

Except , each natural number is a successor of a natural number.

Any set of natural numbers which contains and the successor of every natural number whenever it contains is the set of natural numbers.
These axioms completely define the set of natural numbers . Starting with this, the set of integers is defined in order to make subtraction possible (in the solution of equations of the type , where ). Again, to make division possible (in the solution of equations of the type , where ), the set of rational numbers is defined, and to fill in the gaps in certain representations or classifications, the concept of irrational numbers is introduced. The totality of the rational and irrational numbers is the set of real numbers . Consequently, .