The natural numbers and the Peano axioms

Summary

This lesson defines the natural numbers by means of the five Peano axioms: a set $\mathbb{N}$ with an element $1$ and a successor function $S$ from $\mathbb{N}$ to $\mathbb{N}$ , injective and never reaching $1$ , in which every subset that contains $1$ and the successor of each of its elements is the whole set. It gives names to the successive successors of $1$ , derives the principle of induction for properties from the fifth axiom and applies it, first, to a property of the successor and, then, to problems of increasing difficulty: the sum of the first natural numbers, an inequality that can only be proved after strengthening it, Bernoulli’s inequality, the covering of a board with L-shaped pieces and the commutativity of addition.

Learning objectives

State the Peano axioms and point out which property of the natural numbers each one guarantees.
Write the first natural numbers as iterated successors of $1$ and pass from the symbolic notation to the usual one.
Apply the principle of induction in a simple proof.

The Peano axioms for the natural numbers

The natural numbers, also called positive integers, are the ones we use to count and to measure. They arise in the most natural way in the operation of counting, the simplest in arithmetic. They are defined by the Peano axioms: one assumes that there exists a set, denoted by $\mathbb{N}$ , whose elements are called natural numbers, with a distinguished element, $1$ , and a function $S$ , called the successor, which assigns to each natural number $n$ another natural number $S(n)$ , and that the following five axioms hold. The notions of set, function and injective function that they involve are defined in the lesson on sets, relations and functions.

$1$ is a natural number: $1 \in \mathbb{N}$ .
$S$ is a function from $\mathbb{N}$ to $\mathbb{N}$ : if $n$ is a natural number, its successor $S(n)$ is one as well.
$1$ is not the successor of any natural number: $S(n) \neq 1$ for every $n$ in $\mathbb{N}$ .
$S$ is injective: if $S(n) = S(m)$ , then $n = m$ .
If a set $A$ of natural numbers contains $1$ and, for every $k$ in $A$ , also contains $S(k)$ , then $A$ is the set of all natural numbers: $A = \mathbb{N}$ .

In studying the Peano axioms we notice that the symbol $1$ is only a representation that points to a specific natural number: the one that satisfies these properties. Just as $1$ represents the first natural number, we use familiar symbols to represent its successors:

2 = S(1), \quad 3 = S(2), \quad 4 = S(3), \quad \ldots

In this way, the symbols $2, 3, 4, \ldots$ are abstract entities that represent the successive successors of $1$ . The collection formed by $1$ and all of them is the set of natural numbers:

\mathbb{N} = \{1, 2, 3, 4, \ldots\}

It is also said that the natural numbers are arranged in a sequence, the sequence of natural numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \ldots