Day 41, MAT115

• Infinity is NaN: Not a Number

• Infinity is NaS: Not a Size, either! The reason is that there is more than one size of infinity! This is the most important thing I want you to remember. (This fact will allow you to win on the playground!;)

• The natural numbers are infinite in number ().

• Cardinality

  Two sets have the same cardinality if there is a one-to-one correspondence between them.

  Intuitively: two sets are the same size if they have the same cardinality.

• A sensible rule you can be sure of: a subset is never bigger in size than the set itself.

  And if the sets are finite, the proper subset is always smaller....
  but if the set is infinite, we may actually be able to throw away elements of a set and not change the size of the set!

• The evens, the odds, the rational numbers -- all are the same size as the natural numbers.

• But Cantor showed that the real numbers are larger than the naturals -- because we couldn't find a one-to-one correspondence between them.

  In fact, we proved that it's impossible to have a one-to-one correspondence (by contradiction: we assumed that there was such a correspondence, then showed that that led to an impossibility).

There are infinitely many sizes of infinity. It turns out that the power set of a set is always of larger cardinality than the set itself:

is the set of all subsets of a set. So let's look at an example.

If we have four colors, RGBCyan, then how many subsets of color are there?

# of elements	# of subsets with that # of elements	different subsets
0	1
1	4
2	6
3	4
4	1

What do you notice about the number of sets of each type?

Every finite set is smaller than its power set.

It turns out that every infinite set is also smaller than its power set (which is itself an infinite set, which is smaller than its power set, etc....

The proof is by contradiction, and very similar to Cantor's diagonalization proof.

Let's assume that there's a one-to-one correspondence between an infinite set and its power set. So each element of has a partner subset of .

We construct a new subset of (call it ) that doesn't have a partner. For each and its partner let contain if doesn't (and not contain if does).

Then clearly is different from every other set at the dance -- it doesn't have a partner! The set of subsets is too big for the set itself.

So this shows that there is an infinite string of ever-larger infinities....