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34.4 1-to-1 Correspondence, Transfinite Arithmetic

In the Section 34.3 you learned about the infinite number ℵ0. ℵ0 is the size of the set of natural numbers, N.

In this section we learn how to do arithmetic with ℵ0.

For example, what is ℵ0+1?

0+1 does not mean we add the number 1 to the set of natural numbers.

0+1 does not mean we add the number 1 to the set of natural numbers. 1 is already in N.

So if we put 1 in there again, {1, 0, 1, 2, 3 …}, we just get N again: {0, 1, 2, 3 …}. Remember, repetitions don’t matter.

What we need in order to represent ℵ0+1 is a set with ℵ0 members plus one additional member, that isn’t already in the set.

And what we want to know is what the cardinality of that new set is.

We will use N for our set of size ℵ0, and for the additional member, we’ll pick something that isn’t a number at all.

Alberto is not a number!

How about our friend Alberto. He’s not a number.

Thus the set {Alberto, 0, 1, 2, 3 …} is ℵ0+1.

Here’s how we can prove that ℵ0+1 = ℵ0.

Here’s how we can prove that ℵ0+1 = ℵ0.

We show that there exists a 1-to-1 correspondence between the set {Alberto, 0, 1, 2, 3 …} and N.

If there is a 1-to-1 correspondence, then they must be the same size. Since N is ℵ0, then so is {Alberto, 0, 1, 2, 3 …}.

Here’s the correspondence:

0—Alberto
1—0
2—1
3—2
etc.

In the first column are the elements of N. In the second column are the elements of the set sized ℵ0+1.

Notice that it is essential that we don’t pair every natural number from the first set with that same number from the second set.

Then Alberto would not be paired with anything, and it wouldn’t be a 1-to-1 correspondence.

If you tried that method, and failed, you might have thought that there is no 1-to-1 correspondence.

Just because some attempt at a 1-to-1 correspondence fails, that does not mean that none exists.

But that would be wrong: just because some attempt at a 1-to-1 correspondence fails, that does not mean that none exists.

That would be like inferring that there is no proof of an argument just because our attempt at a proof failed. That would be a bad inference!

But still, you might be thinking, the set with Alberto must be bigger.

Infinite numbers are fun and surprising.

The set with Alberto has “more” in it, since it has all the members of N plus Alberto too! So it has to be bigger.

But that is wrong: infinite numbers are fun and surprising.

What we have discovered is that even though a set “has more in it” in a sense, it could still be the same size as N.

Now you try.

The method we used to show that ℵ0+1 = ℵ0 can be used to show that ℵ0+ any finite number is ℵ0.

For example, take ℵ0+5. We just need five elements that are not in N.

Let’s use the Avengers:

0—Iron Man
1—Wasp
2—the Hulk
3—Thor
4—Ant-Man
5—0
6—1
etc.

That pairing shows that ℵ0+5 = ℵ0.

That pairing shows that ℵ0+5 = ℵ0.

What about ℵ0+ℵ0? We discuss that in the next chapter!