These 5 axioms define the set of natural numbers:

  1. Zero is a natural number.
  2. Every natural number n has a successor S(n), and the successor of any natural number is a natural number.
  3. No two numbers have the same successor.
  4. Zero is not the successor of any natural number.
  5. For any property P, if zero has P and also the successor of any number that has P has P, then every number has P.

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