The peano axioms or the peano postulates, are a set of axioms for the natural numbers.
Within these axioms, there are three kinds of statement:
    The first four axioms are statements about equality
    The second four axioms are about the succession operator, S()
    The ninth, and last axiom, is a second order statement, defining the principle of mathematical induction

They are, in order:
    1. Equality is reflexive. For every natural number x, x = x.
    2. Equality is symmetric. For all natural numbers x and y, if x = y, then y = x.
    3. Equality is transitive. For all natural numbers x, y and z, if x = y and y = z, then x = z.
    4. The natural numbers are closed under equality. For all a and b, if a is a natural number and a = b, then b is also a natural number.
    5. 0 is a natural number.
    6. For every natural number n, S(n) is a natural number.
    7. For every natural number n, S(n) = 0 is False.
    8. For all natural numbers m and n, if S(m) = S(n), then m = n.
    9. If K is a set such that:
      * 0 is in K, and
      * for every natural number n, if n is in K, then S(n) is in K,
    then K contains every natural number.

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