Inifinities.....

...come in all shapes and sizes.

We all know what infinity is, don't we? Or do we?

Sure we do! It's where something goes on forever, never stopping, like the number of numbers. There is no highest number because they go on forever, right? And the decimal portion of Pi goes on forever, does it not? It begins 3.1415926... and never stops and never repeats.

We are so used to this concept of "infinity" that we are almost blase about it.

Allow enlightenment to take place.....

Suppose there is a hotel with an inifinite number of rooms, all occupied by guests.

What does the owner do when a new guest arrives? Why, he puts guest 1 in room 2, guest 2 in room 3, guest 3 in room 4....and so on up to infinity, leaving room 1 empty.

What does the owner do when an infinite number of new guests arrive? This one requires a little more thought.....

Simple, really. He puts guest 1 in room 2, guest 2 in room 4, guest 3 in room 6 and so on up to infinity, leaving an infinite number of "holes" ie vacant odd-numbered rooms. All guests are now occupying rooms. Does that mean the owner somehow now has 2 times an inifinite number of rooms? NO! he still has an infinite number of rooms with a guest in each!

It gets even stranger. Georg Cantor proved that there is more than one "type" of infinity. This guy was involved in set theory, the theory of certain classes of objects, such as the set of cars, the set of brown cars, the set of brown cars with at least 2 wheels, etc.

Sometimes sets can be inside each other, sometimes not. For instance, the set of Green Cars is different from the set of Brown cars, but above the third set "the set of brown cars with at least 2 wheels" is contained inside set 2 "the set of brown cars" AND set 1 "the set of cars".

Cantor took this a little further and argued that sets sometimes can contain themselves. For instance, the set of cars is not a car itself, but the set of all sets IS a set, so it is contained within itself. These are called "self-swallowing" sets. Another example is the set of ALL things that are not brown cars.

So, You have ordinary sets eg "the set of brown cars" which are not members of themselves, and self-swallowing sets eg "the set of all sets".

The question arises: Is the set of all ordinary sets ordinary or self-swallowing?

Inspection shows it is neither. Try it! Assume it is an ordinary set.

So, the set of all ordinary sets belongs to neither group.

The argument put forward above is called the "diagonal" argument for reasons which are not at all clear. Here comes Cantor's same argument, but in the form he proposed.

To begin, suppose we wish to count all numbers. Not just integers such as 1, 2, 3 etc, but ALL numbers, including those with decimals. Well, one of them is pi = 3.1415926.... Another is e = 2.718281828....... and ALL the others in between and around , such 9.80665, and 5.5, 100.99999 and 2.9979. Suppose we list them all in order. For simplicity, let's just consider all those less than 1.

It would look something like

0.1111111111111111....>>>> to infinity
0.2222222222222222....>>>> to infinity
0.3333333333333333....>>>> to infinity....

and suppose our table lists them ALL, one after the other. Obviously there are an infinite number. There is a proof that there is at least one number NOT in our table of the infinitely many "numbers".

It goes like this:

Take the first number, 0.1111111111111.....and change the first column to a "2" creating 0.2111111111111....

Take the second number, 0.222222222222.... and change the SECOND column to a "3" creating 0.2322222222222....

Do the same with the third number. So we will have

0.2111111111111111....>>>> to infinity
0.2322222222222222....>>>> to infinity
0.3343333333333333....>>>> to infinity....

where I have changed the third column in the third number to a "4". Now look at the diagonal. It says 0.234.......to infinity.

It is clear that this number is NOT in our list, as the first column is different from number 1, the second is different for number 2, and so on up to infinity. For any "n", the digit in column n is different from the nth digit of the number "n", so our "made-up" number, derived from adding 1 to the diagonal is absolutely NOT in the list.

But didn't we say that our list was of ALL the numbers all the way to infinity?

The reasoning is that there MUST be another "kind" of inifinity somehow "bigger" than the one we had. In fact, there is not only "one" different kind of infinity, but an infinite many of them, each, in some abstract sense, "bigger" than the last.

This result is true and has had many implications for math, especially for that man Godel again, who said that not everything is proveable within the system. This is a paraphrase of that theorem.