**Axiom**: Peano Axioms (Peano)

Let \(N\) be a set fulfilling the following axioms:

**P1**: \(N\) contains the element \(0\).**P2**: For each element \(n\in N\) there exists a unique element \(n^+\), the so-called**successor of n**^{1}.**P3**: There is no element \(n\in N\) such that \(n^+=0\) (i.e. \(0\) is not a successor of any element of \(N\)).**P4**: If two elements \(n,~m\in N\) have the same successors \(n^+=m^+\) then they are the same \(n=m\).**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*

## 1.**Definition**: Set of Natural Numbers (Peano)

## 2.**Explanation**: Why do the Peano axioms define natural numbers?

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