Peano postulates

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Giuseppe Peano proposed the following 5 axioms for the natural numbers; they have come to be known as the Peano postulates.

  • There is a natural number 0.
  • Every natural number a has a successor, denoted by a + 1.
  • There is no natural number whose successor is 0.
  • Distinct natural numbers have distinct successors: if a <> b, then a + 1 <> b + 1.
  • If a property is possessed by 0 and and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.

These axioms are sometimes paraphrased differently, starting at 1 instead of 0.