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The natural numbers

Definition 6.1.1 (Peano's axioms)   We denote by $ \mathbb{N}$ any set for which the following properties hold:
(Pe1)
There exists an injection $ \phi: \mathbb{N}\longrightarrow \mathbb{N}$. For $ x \in \mathbb{N}$, $ \phi (x)$ is called the successor of $ x$.
(Pe2)
There exists an element, called zero and denoted by 0, which has no pre-image by $ \phi$.
(Pe3)
Principle of Induction: If a subset of $ \mathbb{N}$ contains 0 and the successor of each of its elements, then this subset is equal to $ \mathbb{N}$.
An element of $ \mathbb{N}$ will be called a natural number.

Among Mathematicians, there are people who replace the second axiom by ``there exists an element, called one and denoted by $ 1$ `` ... . The set they construct is in bijection with the set we construct it. Nevertheless, we prefer the rpesent construction as it introduces the number 0 with the same civil rights as any other natural number. An important occurence is in Theorem 1.28. There exist situations in which two of the above axioms hold, but not the third. For example, let $ f: \mathbb{N}\longrightarrow \mathbb{N}$ be such that $ \phi (x)=3^x$. The first and second axioms hold but not the third one. Let us elaborate on the axioms:
  1. As $ \phi$ is an injection, two different natural numbers have different successors.
  2. Denote $ x'=\phi (x)$; then the first axiom says:

    $\displaystyle \forall x \in \mathbb{N}, \; \forall y \in \mathbb{N}, \;
 x \neq y \Longrightarrow x' \neq y'.$    

  3. In order to be consequent with the intuition (and the ordinary notations), we have: $ 0'=1$, $ 1'=2$, etc.
  4. There exists no element $ x$ such that $ x'=0$.
  5. The principle of induction says that if a property is true for 0 and is hereditary, i.e. passes from a number to its successor, than this property is true for all the natural numbers. For example, people can infect one the other with some illness, but if nobody suffers from this affection, then the epidemy has no possibility to begin.

Proposition 6.1.2   Let $ P$ be a property of natural numbers. Assume that if $ P$ holds for a natural number $ n$, the it holds for $ n'$. Then $ P$ is true for all the natural numbers whenever it holds for 0.

The Induction Principle and this last proposition say that we must avoid a frequent mistake: to verify only that the given property is hereditary, forgetting to verify that it holds for 0.

Definition 6.1.3   If $ n$ and $ m$ are two natural numbers and $ n=m'$, then $ m$ is called the predecessor of $ n$.

For example, 2 is the predecessor of $ 3$.

Proposition 6.1.4   Every natural number different from 0 has a unique predecessor.

Proof. Denote by $ M$ the set of natural numbers for which the folloing property $ P$ holds:
$ P(n)=$''either the number $ n$ is equal to 0 or it has a predecessor''
. By the Induction Principle, $ M=\mathbb{N}$, i.e. every natural number (0 excepted) has a predecessor. As the mapping $ n \mapsto n'$ is an injection (first Peano axiom), the predecessor of a non zero natural number is unique. $ \qedsymbol$

The following proposition is a direct consequence of Peano's axioms (actually, $ N_0$ acts in place of 0).

Proposition 6.1.5   Assume that some property $ P$ verified the following conditions:
(i)
$ P$ holds for the natural number $ N_0$.
(ii)
If $ P$ holds for a natural number $ n$, then $ P$ holds for $ n'$.

Example 6.1.6   We prove that for any convex polygon with $ n$ vertices, the sum of the internal angles is equal to $ (n-2) \cdot 180^o$.

=
Solution
By induction on $ n \geq 3$.


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root 2002-06-10