DISCRETE MATHEMATICS
W W L CHEN
c
±
W W L Chen, 1982, 2008.
This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990.
It is available free to all individuals, on the understanding that it is not to be used for ﬁnancial gain,
and may be downloaded and/or photocopied, with or without permission from the author.
However, this document may not be kept on any information storage and retrieval system without permission
from the author, unless such system is not accessible to any individuals other than its owners.
Chapter 3
THE NATURAL NUMBERS
3.1. Introduction
The set of natural numbers is usually given by
N
=
{
1
,
2
,
3
,...
}
.
However, this deﬁnition does not bring out some of the main properties of the set
N
in a natural way.
The following more complicated deﬁnition is therefore sometimes preferred.
Definition.
The set
N
of all natural numbers is deﬁned by the following four conditions:
(N1) 1
∈
N
.
(N2) If
n
∈
N
, then the number
n
+ 1, called the successor of
n
, also belongs to
N
.
(N3) Every
n
∈
N
other than 1 is the successor of some number in
N
.
(WO) Every nonempty subset of
N
has a least element.
The condition (WO) is called the Wellordering principle.
To explain the signiﬁcance of each of these four requirements, note that the conditions (N1) and
(N2) together imply that
N
contains 1
,
2
,
3
,....
However, these two conditions alone are insuﬃcient to
exclude from
N
numbers such as 5
.
5. Now, if
N
contained 5
.
5, then by condition (N3),
N
must also
contain 4
.
5
,
3
.
5
,
2
.
5
,
1
.
5
,
0
.
5
,

0
.
5
,

1
.
5
,

2
.
5
,...,
and so would not have a least element. We therefore
exclude this possibility by stipulating that
N
has a least element. This is achieved by the condition
(WO).
Chapter 3 : The Natural Numbers
page 1 of 6