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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 \(n^+\).