Introduction

The Set $\mathbb{N}$ of Natural Numbers

We denote the set ${ 1, 2, 3}$ of all positive integers by $\mathbb{N}$.

The following properties of $\mathbb{N}$ are obvious:

• N1. $1$ belongs to $\mathbb{N}$.
• N2. If $n$ belongs to $\mathbb{N}$, then its successors $n+1$ belongs to $\mathbb{N}$.
• N3. $1$ is not the successor of any element in $\mathbb{N}$.
• N4. If $n$ and $m$ in $\mathbb{N}$ have the same successor, then $n=m$.
• N5. A subset of $\mathbb{N}$ which contains $1$, and which contains $n+1$ whenever it contains $n$, must equal $\mathbb{N}$.

Properties N1 through N5 are known as the Peano Axioms.

Axiom N5 is the basis of mathematical induction.

Let $P_1, P_2, P_3\ ...$ be a list of statements or propositions that may or may not be true. The principle of mathematical induction asserts all the statements $P_1, P_2, P_3, ...$ are true provided

1. $P_1$ is true,
2. $P_{n+1}$ is true whenever $P_n$ is true.