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Peano Arithmetic

• Theory of natural numbers.
  • Object constant 0 (zero).
  • Unary function constant ' (successor).
• Axioms
  1. 0 is not the successor of any natural number:

     forall x: x' != 0.

  2. Different natural numbers have different successors:

     forall x, y: x' = y' => x = y.

  3. F holds for every number, if F holds for 0 and with every number also for its successor:

     (F[x <- 0] /\  (forall x: F => F[x <- x+1])) => forall x: F.