Peano Axioms

  1. 1 is a natural number.
  2. Each natural number n has a successor n + 1.
  3. Two natural numbers are equal if their successors are equal.
  4. Except 1, each natural number is a successor of a natural number.
  5. Any set of natural numbers which contains 1 and the successor of every natural number k whenever it contains k is the set \mathbb{N} of natural numbers.
These axioms completely define the set of natural numbers \mathbb{N}. Starting with this, the set \mathbb{N} of integers \mathbb{Z} is defined in order to make subtraction possible (in the solution of equations of the type x + n = m, where n,m \in \mathbb{N}). Again, to make division possible (in the solution of equations of the type nx = m, where n \in \mathbb{N} \wedge m \in \mathbb{Z}), the set of rational numbers \mathbb{Q} 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 \mathbb{R}. Consequently, \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}.