Peano's Axioms

1. N is a set and 1 ∈ N.
2. Each element x of N has a unique successor in N denoted x'.
3. 1 is not the successor of any element of N.
4. If x' = y' then x = y.
5. If M ⊂ N satisfies both
   • 1 ∈ M
   • x ∈ M implies x' ∈ M
   then M = N.

Named for Giuseppe Peano, who published them in 1889, these axioms define the system of natural numbers. We think of this set as a string of beads, beginning with 1 and stretching infinitely towards the right. It is an interesting exercise to delete one axiom at a time, and see what non-equivalent pictures result.

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