Set
Definition of Natural Numbers
Subjects to be Learned
- The set of natural numbers
Contents
The set of natural numbers can be defined using set theory.
For this definition only set theory and logic are required.
Definition: the set of natural numbers N
is the set that satisfies the following:
- Basis Clause:
N
- Inductive Clause:
If n
N ,
then
n
{ n }
N.
- Extremal Clause: If S
N and S satisfies 1 and 2 ,
then S = N
According to this definition, the set of natural numbers has as its elements
the sets
,
{
} ,
{
,
{
} } ,
{ ,
{
} ,
{
,
{
} }
} , ... .
These are denoted as 0, 1, 2, 3, ... .
Note that the definition of N given as an example of
representation of set
and
recursive
definition
is not acceptable, because it uses the addition of natural numbers which requires
the concept of natural number. Also it uses 0 without defining it.
The definition given here, on the other hand, relies only on the assumptions on set
and logic.
Intuitively we can view these numbers as follows: first the empty set is a set
that has nothing in it. Therefore it agrees with our intuition of number 0.
We now have one concept which is the concept of zero. Thus construct a set that has
0 in it. It contains one object 0. That is the number
1. Now we have two distinct objects:
0 and 1. So construct a set that has these two objects.
that is the number 2. Continuing in this manner, we can see the correspondence
between the natural numbers and these sets.
This way we can characterize numbers without relying on anything except the concept
of set and logic.
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