Natural Numbers and Integers
- Axiom of Induction
Suppose \(S\) is a subset of \(\mathbb{N}\) that satisfies
- \(0 \in S\)
- If \(n\in S\), then \(n+1 \in S\)
Then \(S = \mathbb{N}\)
The addition here is defined by “successor”. This axiom is the principle of the commonly used mathematical induction.
With axiom of induction, we can prove least-number principle and greatest number principle.
- Least-Number Principle
- Suppose \(T\) is a non-empty subset of \(\mathbb{N}\), then, \(\exists t_{0}\in T\) such that \(\forall t\in T, t_{0}\le t\).
- Greatest-Number Principle
- Suppose \(M\) is a non-empty subset of \(\mathbb{N}\). If \(M\) has an upperbound, i.e., \(\exists a \in \mathbb{N}\) such that \(\forall m\in M, m \le a\). Then, there \(\exists m_{0}\in M\) such that \(\forall m\in M, m\le m_{0}\).