This exposition is going to be purely mathematical. Any connection to the real world is completely
accidental.
In mathematics, infinite sets abound. For example, we investigate the set of counting numbers,
N={1,2,3,4....}
the extension of this, the set of integers
Z={...,-3,-2,-1,0,1,2,3,...}
and the set of real numbers, R, which consists of all numbers represented by decimals, like
pi=3.1415926535.....
or
sqrt(2)=1.41421356237309....
or even
1/3=.3333333....
In each case, the all numbers with the stated property is an infinite set. One question that
arises is how to compare two infinite sets. The usual way of doing so is by the concept of
cardinality. We say two sets are of the same cardinality if it is possible to 'pair off'
the two sets so that each element of either is paired with exactly one element of the other.
But this concept leads to some rather counter-intuitive results. For example, the sets N and Z
above are 'the same size' if we use cardinality:
N <-> Z
1 <-> 0
2 <-> 1
3 <-> -1
4 <-> 2
5 <-> -2
6 <-> 3
7 <-> -3
....
....
....
In this way, every counting number on the left is paried with eactly one integer on the right
and vice versa. Because this is possible, we have that N and Z have the same cardinality.
Of course, you might say, both are infinite, so they *must* be the same size, right? Well, no.
It turns out that it is impossible to pair off every counting number with every decimal number.
This was discovered by Cantor and the argument we use is called the diagonalization argument.
We restrict to just the decimal numbers between 0 and 1 and we suppose that there is, in fact,
such a pairing that gets *every* decimal between 0 and 1. Let's write out this pariing
N <-> R
1 <-> .a_1 a_2 a_3 a_4 .....
2 <-> .b_1 b_2 b_3 b_4 .....
3 <-> .c_1 c_2 c_3 c_4 .....
4 <-> .d_1 d_2 d_3 d_4 .....
...
...
...
Here, each line pairs some counting number with a decimal. For example, we might
have
1<-> .2349402470925....
in whch case a_1 =2, a_2 =3, a_4 =4, a_5 =9, etc. The subscripts are just to keep track of where we are
in the expansion.
But, I'm going to show how to get a decimal that is NOT on this list (showing a complete list is impossible).
Let A_1 be any numeral that is not 0, 9, or a_1
Let A_2 be any numbral that is not 0, 9, or b_2
Let A_3 be any numeral that is not 0, 9, or c_3
....
In essence, we go down the diagonal of our list, making sure we pick a digit different than the one on the diagonal
(as well as 0 and 9). We then consider the decimal number
.A_1 A_2 A_3...
This cannot be the first number on our list because the first decimal place isn't a_1.
It cannot be the second number on our list because the second decimal place isn't b_2
It cannot be the third number on our list because the third decimal place is not c_3
.....
In other words, it cannot be *anything* on the list! No such list can be complete!
This shows that the set of decimal numbers, R is actually a *larger* infinity in size than N.
Now, we can actually do more. If we have any set at all, say X, we can consisder the collection P(X) of all subsets
of X. This new collection is always a larger set (in terms of cardinality) than the orginal set X. It turns out
that R and P(N) have the same cardinality.
But this means, we can consider the sequence of ever larger cardinalities N, R=P(N), P(R), P(P(R)), ....
This means there are an infinite number of distinct sizes of infinity!!!
But we can go further. We can ask if there is a cardinality larger than all of these. And the answer is yes.
All we have to do to get one is put all of the sets in our list together into one bigger set. The cardinality of
this set has an infinite number of cardinalities before it. But, if you look carefully, you can see that the number
of such smaller ones is the same size as N.
It turns out that in mathematics, there are cardinalities with the same number of precursors as the number of elements of R,
or of P(R), or of P(P(R)), etc.
The kicker: in math, these are all known as 'small cardinals'. Large cardinals are much, much larger than any we have seen here.
