WITHOUT NUMBERS (Part 2 - Peano)

WITHOUT NUMBERS (Part 2 - Peano's Axioms)

For Part 1-WITHOUT NUMBERS (Part 1- The horror)

Thus ended the narrative of the mathematics lecturer who had come as substitute for a day. The professor managed to catch the attention of the class in the dreadful afternoon hour.  There was pin drop silence that followed his story. He went on to say that we should live the story we want to tell.  This is the story of numbers and how mathematics evolves its structure.

We were all like


He went on to say that Mathematics, in its essence, is a subject in which one begins with a set of concepts and rules and then rigorously works out their precise consequence.

Thus began an interesting class..

First of all I am sure you all have the same question
 What are natural numbers then?

The informal definition your textbooks  will give 
"A Natural number is any element of set N= {0,1,2,3....} ( We often call the set containing zero and natural numbers as whole numbers. But it is fine here)
which is the set of numbers created by starting with 0 and counting forward indefinitely"

Before we could note that down.

Our Professor...





This definition doesn't seem so satisfactory because then the question is what is counting forward indefinitely?

Intuitively the definition is okay, but we all knew from the story (see Part 1-WITHOUT NUMBERS (Part 1- The horror) our intuitions are also not entirely acceptable; for instance how do we know it is possible to keep counting indefinitely without cycling back to 0? More importantly how do we perform operations like addition, Multiplication, exponentiation and so on ?

One standard way of doing it is in terms of Peano's Axioms. The idea is to just present a complete set of axioms to get N 

and Nothing is taken for granted or treated as obvious.

The intuitive picture is N consists of 0 and everything can be obtained from 0 by incrementing. So our beginning is in the existence  of 0 and an operation called incrementing.

Axiom 1: There exists a natural number denoted by 0
We can imagine emptiness or nothingness. We denote 0' the successor of 0 and 0'' as successor of 0' that is successor of successor of 0. We also use usual notation 1=0', 2=0'',...

Axiom 2 : If n is a natural number then n' is also a natural number
0 is a natural number we know from axiom 1, 
0' (1) is a natural number by axiom 2
0'' (2) is a natural number since 1 is a natural number
0''' (3) is a natural number since 2 is a natural number

Professor stopped and asked us to consider if these two Axioms are enough to define N?

This time we did identify the problem that is the above definitions do not rule out the possibility of 0 being a successor of a natural number.
Ex; consider the set{0,1,2} where 0'=1, 1'=2 , 2'=0. It does satisfy axiom 1 and 2 

Quick fix 
Axiom 3 : 0 is NOT the successor of any natural number

Oh we should be done now...

But consider {0,1,2,3} and 0'=1, 1'= 2, 2'= 3, 3'=1 

The successor should not wrap back to the earlier natural number

He was definitely pushing us to the limit where we all did go like..



 
Then we need

Axiom4: If n, m are natural number such that n'=m', then n=m or equivalently if  (the symbol != will be used to represent "not equal to") n != m, then n' != m'. This means different natural numbers must have different successors.

ex Prove 5 != 2  ( 0''''' != 0'')
if       5=2
---->  4'= 1'
---->  4= 1 ( by Axiom 4)
---->  3'=0'
---->  3 = 0 ( by Axiom 4)
---->  2' = 0
wait a minute that is not possible because it would then be a contradiction to Axiom 2 and 3

Hence 5 cannot be equal to 2

Our lecturer therefore making it the best class in long time concluded
These four axioms seems to give us all the natural numbers distinct from each other.

While we were all satisfied this one student didn't seem particularly convinced it is complete and said that these axioms 1 to 4 do gives 1,2,3,4 ... in set N but none of these axioms rule out the possibility of the other elements which we do not want example {0, 0.5, 1, 1.4,...}

The lecturer look stunned at the boy's observation and tell us all

It is true these Axioms do not rule out the possibility of entirely different natural numbers represented by 0.5 and then satisfying axiom 1 to 4 such that
0=0.5, 1.5=(0.5)', 2.5 = (1.5)' ... By now you have realised 1, 2, 3 are just symbols. Our interest lies in its structure.

To rule out this possibility, a very ingenious axiom, not very intuitive immediately, axiom is formulated

Axiom 5: (Based  on principle of mathematical induction) 
Let P(n) be a "property" pertaining to a natural number n. Suppose that P(0) is true and suppose whenever P(n) is true, P(n') is also true. Then P(n) is true for all n.

The Axiom 5 is in fact very very deep and really ingeneous.
 The term property looks a little vague but think of P(n) being related to property " n is even"  , n is equal to 3. Ofcourse these examples are still vague as we have not defined concepts yet. But once we define we rule out the possibility of other elements creeping in.

These are in fact Peano's five Axioms.
Thus the hour came an end and we had a wonderful math hour. The substitute lecturer was impressed with that one student's insight and asked him his name.

The student said " Joiston Reich sir"

That was the last we saw of our substitute lecturer.

Check them out (These were the links of inspiration)
A day without math 

Lots of young kid's stories on how their day would be without math
http://www.globalclassroom.org/nonumb.html


Books referred 

• Analysis 1, 2 by Terence Tao, Hindustan Book Agency, 2006
• Claude W Burril, Foundations of real number, McGraw-Hill(1967)

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