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

1.Definition: Set of Natural Numbers (Peano) 

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