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## Axiom: Peano Axioms (Peano)

Let $$N$$ be a set fulfilling the following axioms:

1. P1: $$N$$ contains the element $$0$$.
2. P2: For each element $$n\in N$$ there exists a unique element $$n^+$$, the so-called successor of n1.
3. P3: There is no element $$n\in N$$ such that $$n^+=0$$ (i.e. $$0$$ is not a successor of any element of $$N$$).
4. P4: If two elements $$n,~m\in N$$ have the same successors $$n^+=m^+$$ then they are the same $$n=m$$.
5. P5: If a subset $$A\subset N$$ contains the element $$0$$ and with each element $$n$$ contained in it it also contains the successor $$n^+$$, then $$A$$ must be the set $$N$$ (principle of induction).

1 In P2 we denote $$n$$ the predecessor of $$n^+$$.

| | | | | created: 2014-03-06 13:38:09 | modified: 2014-06-21 14:20:57 | by: bookofproofs