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Peano Axioms
Let N be the set of natural numbers. Then the properties satisfied by N, known as the Peano’s Axioms, are:

Axiom 1: 1 N, i.e. 1 is a natural number.

Axiom 2: For each n N, there exists a unique natural number n* N called the successor of n.

Axiom 3: 1 ≠ n*, n N, i.e. 1 is not the successor of any natural number.

Axiom 4: ∀ m, n N, m* = n* m = n, i.e. each natural number, if it is a successor, it is the successor of a unique natural number.

Axiom 5: Principle of finite induction (P.F.I.)

If S ⊂ N be such that

(i) 1 S and

(ii) m S m* S, then

S = N.

Note:
Axiom 1 assures that N is not a null set, i.e. N ≠ Ø.

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