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# The natural numbers

Definition 6.1.1 (Peano's axioms)   We denote by any set for which the following properties hold:
(Pe1)
There exists an injection . For , is called the successor of .
(Pe2)
There exists an element, called zero and denoted by 0, which has no pre-image by .
(Pe3)
Principle of Induction: If a subset of contains 0 and the successor of each of its elements, then this subset is equal to .
An element of 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  ... . 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 be such that . The first and second axioms hold but not the third one. Let us elaborate on the axioms:
1. As is an injection, two different natural numbers have different successors.
2. Denote ; then the first axiom says:

3. In order to be consequent with the intuition (and the ordinary notations), we have: , , etc.
4. There exists no element such that .
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 be a property of natural numbers. Assume that if holds for a natural number , the it holds for . Then 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 and are two natural numbers and , then is called the predecessor of .

For example, 2 is the predecessor of .

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

Proof. Denote by the set of natural numbers for which the folloing property holds:
''either the number is equal to 0 or it has a predecessor''
.
• The fact that is obvious.
• Suppose that some natural number belongs to . Then has a predecessor, thus .
By the Induction Principle, , i.e. every natural number (0 excepted) has a predecessor. As the mapping is an injection (first Peano axiom), the predecessor of a non zero natural number is unique.

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

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

Example 6.1.6   We prove that for any convex polygon with vertices, the sum of the internal angles is equal to .

=
Solution
By induction on .
• For , the polygon is a triangle; the sum of its angles is .
• Suppose that for some natural number , the desired property holds. Consider a convex polygon with vertices and denote the vertices clockwise by , , , . By the induction hypothesis, the sum of the angles of the convex polygon is equal to ; moreover, the sum of the angles of the triangle is . Thus, the sum of the angles of the polygon is .

Subsections

Next: Addition Up: The classical sets of Previous: The classical sets of   Contents
root 2002-06-10