Peano’s Axiom

Posted on July 5, 2017 by Sumant Sumant

Following are the 5 Axioms of Peano or also called as Peano’s Postulates
N1. 1 \in \mathbb{N}
N2. If n \in \mathbb{N} \Rightarrow n+1 \in \mathbb{N}
N3. 1 is not the successor of any element in \mathbb{N}.
N4. If two numbers m,n \in \mathbb{N} have the same successor then m=n
N5. A subset of \mathbb{N} which contains 1, and which contains n+1 whenever it contains n, must equal \mathbb{N}

Q. What is the significance of Peano’s Axiom ?
Most familiar properties of \mathbb{N} can be proved using Peano’s Axioms.

Q. How do you prove N5 ?
Given the set contains 1. If
We will prove by Contradiction
Suppose there is a set S \subseteq \mathbb{N} and S \ne \mathbb{N}, that means there is a smallest element n_0 \in \{n\in \mathbb{N}| n \not \in S \}. Obviously n_0 \ne 1 as 1 \in S. As n_0 is the smallest element which is not in S \Rightarrow n_0-1 \in S. But if n_0-1 \in S \Rightarrow n_0-1+1=n_0 \in S so we have a \Rightarrow \Leftarrow and our assumption that there exists a number outside set S is False and S=\mathbb{N}

About Sumant Sumant

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