1.  is a natural number.
2. Each natural number  has a successor.
3. Two natural numbers are equal if their successors are equal.
4. Except  each natural number is a successor of a natural number.
5. 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 up 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,.