Infinity: let's think about it. Let's think about The Infinite hotel which is full up with an occupant in each room. Even though we are full we can easily accommodate infinitely many new guests who all turn up on an infinite bus! We just ask everybody already in room, n say,   to move into room 2n (all the even numbered rooms) . Then as the new folk check in, the kth one goes into room 1+2(k-1) (the odd numbered rooms).

 

Put concisely, infinite sets have the wonderful property that they can be put into one to one correspondence with a proper subset of themselves. The occupants of the rooms in the hotel can be moved into the all the even number rooms - we could also move them back without any ambiguity! So there is a one-to-one correspondence between the set of all rooms and the set of all even numbered rooms - a proper subset.

 

Mathematicians generally didn't like this property, historically. It was one of many problems with thinking about infinity. But one man, Georg Cantor, embraced this property. Indeed he defined infinite sets as those that could be put into one to one correspondence with a proper subset of themselves. He referred to the "cardinality" of infinite sets, rather than the "size" of finite sets. Two sets have the same cardinality if there is a one to one mapping between them.

 

One-to-one mappings (called bijections) are really just a re- labelling of one set of elements with the names of the elements form another set and vive versa. Rename the England football starting line up, with the eleven chosen distinct pet dog's names. Then you can still commentate on the game, using the pet's names, and a listener will know exactly what and who you are talking about! “Fido wins the ball and passes to Rover…”

 

A good exercise to start with is to think about COUNTABLE sets.

 

These are sets of elements that can be counted. They have the same cardinality as the natural; numbers {1,2,3,.}. We say they are countable. We may lay them out in the order in which we cont them. We have seen that the set of even numbers is countable: the mapping n- 2n is a bijection. The odd numbers are countable: n -> 1+2 (n-1).

 

Now for an exciting surprise - the set of all fractions, p/q, where p and q are integers, is countable.

 Lay them out as an infinite array with p/q in the p th column and q th row.

 

 1/1       2/1       3/1       4/1        5/1      .

 

 1/2       2/2       3/2       4/2       5/2…….      .

 

 1/3       2/3       3/3       4/3       5/3       .

 

 1/4       2/4       ¾         4/4       5/4       .

 

 ...............

 

 

 

You have infinitely many rows going down the page and infinitely many columns to the right. Now we "count them" by going over successive diagonoals. Number 1 is 1/1 in the top left. Number 2 and Number 3 are ½ (1st element of second row) and 2/1 (2nd element of 1st row), respectively. Then the next diagonal: Numbers 4,5 and 6 are 1/3, 2/2, 3/1.. And so on. ¼, 2/3, 3/2, 4/1 are Numbers 7, 8, 9 and 10, respectively. After a little work you can demonstrate that p/ q is number p+ (p+q-2)(p+q-1)/2 . Eg 2/3 is Number (2+(2+3-2)(2+3-1)/2)=(2+6)=8.

 

Are all infinites sets Countable? No. There are different Cardinalities. There are some infinite sets which are not countable. For example all of the numbers on the real number line between 0 and 1 cannot be counted.

 

The proof, or practical demonstration, of this requires the idea of a "contradiction".

 

Assume that they can be counted. So let us count them. Assume we have counted them and lay all these real numbers (between 0 and 1) out in a list in the order they have been counted. Represent each in the list by its infinite decimal expansion, adding zeros if this is finite.

 

 There will be a

"Number 1", say,                      0.6543456000…., and

"Number 2", say,                      0.2340000000….; 

"Number 3", say,                     0.8000000000…..;

"Number 4", say,                      0.3333333333….;

and so on.

 

Now comes a trick. Form up a new number, say P, that is deliberately chosen by us so that it differs from "Number N" in the Nth decimal place.

 

So take P=0.5812…               

Notice

the 1st decimal place for P must notmatch the 6 in the first decimal place for our "Number 1" above;

the 2nd place for P must not match the 3 that appears in the second decimal place

for "Number 2" above, ;

the 3^rd place for P must not match the 0 that appears in the third decimal place

 or "Number 3" above ;

the 4th place for P must not match the 3 that appears in the fourth decimal place

 for "Number 4" above; and so on..

 

Now ask if P is in the Counted List. It is a number between 0 and 1 for sure, so we said it must be there. And we can know its decimal expansion. But suppose it is Number K in the list - then we have a problem because P has been chosen to disagree with Number K in the Kth decimal place. So P cannot be in the list. Contradiction! We said that the list must contain all numbers between 0 and 1. So that can't be right therefore those numbers cannot be countable.

 

So the Cardinality of the real numbers between 0 and 1 is different from that of the counting numbers. But the Cardinality of the fractions between 0 and 1 is the same as that of the counting numbers.

 

If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. Cantor's views prevailed and modern mathematics accepts actual infinity.

 

Cantor developed a system of "transfinite numbers", in which the first transfinite cardinal is aleph-null , the cardinality of the set of counting numbers.

 

Just as the set of real numbers between 0 and 1 has a bigger Cardinality than the set of integers, given any set one can construct a new set – called its power set, which is the set of all subsets of that set, which has a greater cardinality. So the ere is an infinite staircase of increasingly large type of infinities. An infinity of infinities with no end.

 

We have also shown above that the cardinality of the real numbers between 0 and 1 is greater than that of the counting numbers (since the counting numbers are in one to one correspondence with a smaller subset of the real numbers - the fractions.)

 

The “continuum hypothesis” states that there is no cardinal number between the cardinality of the real numbers and the cardinality of the natural, counting numbers, numbers. You cannot construct an infinite set which has a cardinality between them. However, this hypothesis can neither be proved nor disproved within the widely accepted terms of mathematics.

 

Oh dear. one to one mappings; apparent paradoxes; the idea proof by contradiction, there is an infinity of different types of infinity, and the notion of things which can never been shown to be true or untrue!