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The "natural numbers" are the numbers we use for counting things. Generally, they're defined as the positive integers, perhaps with the inclusion of zero:

    0, 1, 2, 3, ...

That definition, while it gives the correct idea, is unfortunately rather circular, since the integers are defined in terms of the natural numbers.

More specifically, the "counting numbers" are the positive integers. The "whole numbers" are the nonnegative integers: 0, 1, 2, 3, ... The "natural numbers" can refer to either set.  On this page we will be including 0 in the "natural numbers".

Also called the "finite ordinals", the natural numbers (starting with 0) are also the initial segment of the set of all "ordinal numbers". They are used to count sets of things that are well ordered.

The natural numbers may also be defined as the Peano integers: the set of numbers which satisfy the five Peano axioms:

The Peano Axioms

(P.1)   "0" is a natural number.

(P.2)  There is an operation called "successor".  For every natural number n, successor(n) is also a natural number.

(P.3)  "0" is not the successor of any natural number.

(P.4)   Two natural numbers with equal successors are equal.

(P.5)   [Induction axiom]  A subset of the natural numbers which contains "0" and which contains the successor of every member is equal to the set of natural numbers.

Additional Properties of the Natural Numbers

It is possible to define multiplication, addition, and comparisons using the Peano axioms.  We'll demonstrate that later on this page.

A Basic Model for the Peano Axioms

For a more concrete definition of the natural numbers, as well as a (limited) proof of the consistency of the Peano axioms, we can construct a model of the nonnegative integers. The foundation of the model is the object we will use as zero, along with a "successor" function, which we will call succ. Given any number, the successor function provides us with the next number; we use it to inductively define all the rest of the natural numbers.

   

   


and then:

    

Our model, then, consists of "0" and all finite chains of successors to "0".

Satisfaction of the Peano Axioms

We'll now prove the Peano axioms as theorems within the model.

(P.1) and (P.2) are trivially true.

(P.3) is nearly as straightforward.  Consider any element of the model, a.  Call its successor b.  Then we have:

    

But then b contains at least one element, "a".  So, b is nonempty; so, b ≠ 0, and no natural number can have zero for a successor.

To prove (P.4), recall that every natural number is obtained from 0 by applying succ() a finite number of times.  Furthermore, note that each member is defined as a set and the number of elements in each member is equal to the number represented by that member.  So, 0 contains 0 elements, 1 contains 1 element, 2 contains 2 elements, and so forth.

Now suppose

   

Count the elements in the set c.  Call the number of elements n.  Then we have

   

But then we must have

   

and so

   

and so a and b must be equal.

To prove (P.5), assume it is false.  Consider a subset of the natural numbers, S, which contains 0 and which is closed under the succ() operation, and assume that it does not contain all the natural numbers.   Choose a number which is not in S; call that number x.

Since x is a natural number, it's possible to get from 0 to x with a finite chain of succ operations.  Write the members of the chain down, starting with 0:

    0
    succ(0)
    succ(succ(0))
    succ(succ(succ(0)))
    ...

As you write them down, check each one:  Is it a member of S, or not?  Certainly, 0 is, by assumption.  But some member on the list must not be in S, since, at least, x is not.  Upon encountering the first number which is not a member of S, back up.  Since you stopped at the first one that wasn't in S, the one before it must be in S.  But by assumption, its successor must be in S also; but we thought it wasn't.  So there is a contradiction, and there must be no such number as x.  So, S contains all the natural numbers.

This completes the proof that we really have constructed a model for the Peano axioms.  On the rest of this page, we will refer to this model as .

Additional Definitions:  Comparison, Addition, Multiplication

In the following definitions, we'll just use the properties determined by the Peano axioms, rather than depending on any way on the "internal structure" of the objects in our model.

We define a predecessor function, pred(x), implicitly, in terms of succ(x):

   

We observe, by (P.3) and (P.4), that pred(x) is uniquely defined for all x≠0, but pred(0) is not defined.

We also need the concept of the n^th successor, where  .  We define it inductively:

    

We can then define a comparison operator, "≤", as:

   
 
Addition is straightforward:

   

Multiplication must also be defined inductively.

    
 
The construction of the model is now complete. I've been rather casual about applying the induction axiom, though, so some of the steps I've laid out may seem a bit fuzzy.

The natural numbers are important not just in themselves, but because they are the building blocks from which we can construct a model of the integers, the rational numbers, and the real numbers, upon which most of the rest of mathematics is based.

In these troubled times it's worth remembering that these are properly called the "Arabic numbers", and they were not invented in Europe.

Originally posted on Anarchopedia, in more limited form, on 03:49, 19 Nov 2004 (UTC)



Page created on 11/07/2007