In our very first posts we talked about how big infinity is, and how there is more than one size of infinity — countable infinity, which is infinite but you could count it, and larger, uncountable infinities, which are so large that you cannot count them.
The standard example for countable infinity is the counting numbers (1, 2, 3, etc.). Clearly infinite, but also you can clearly count them. For uncountable infinities, the standard example is all the numbers, say between 0 and 1. This is also clearly infinite, but no matter how you arrange it, you can’t count all of those numbers.^{1}
But the size of infinity, which is measured by directly comparing which things are in which sets, is different than the length of infinity.
We want a normal kind of length — The length of all the points between 0 and 1 should be 1. Similarly, a single point should have zero length.
So, somewhere between a single point and all the points between 0 and 1, we go from length zero to length one. Where did it happen? Two points has no more length than one point, and the same goes for 10 billion points. Similarly, if I take all the numbers between 0 and 1, and take away a single point, the length of all those points should still be 1.
The famous non-mathematical version of this is the sorites (so-RITE-eez) paradox. If you have a heap of sand, and take a single grain away, you still have a heap of sand. But if you keep removing one grain at a time, eventually you will only have a single grain remaining, and that’s clearly not a heap. So when did it stop being a heap?
How do we measure the length of infinity?
As commonly occurs in math, the answer is to carefully define what we mean by “length.”
Let’s start with what we can all agree on — the length of an interval.
An interval is all the numbers between two endpoint numbers. For example, is all the numbers between 0 and 1, including 0 and 1, while is all numbers between -17 and 4, not including -17 and 4.^{2}
Whether or not an interval contains its endpoints, we can all agree that the length of that set should be the right endpoint minus the left endpoint. For example, the length of .
Great, now for the complicated part.
We need to use intervals to define the length of any set of points.^{3} What we’ll do is estimate the length of the set using intervals.
For any set, we can cover it using intervals. For instance, if our set was the single point 0, we could cover it with the interval .
If our set was , we could cover it with the intervals , , and .
Since we’re using intervals, the total lengths of these intervals is easy to calculate. In the first case, the length was 1, and in the second case the total length was .
The intervals we choose contain the set we care about, and so the length of the set should be less than the length of the intervals containing it. So, the length of the single point 0 should be less than 1, and the length of should be less than 7/4.
Of course, we could have picked other intervals to cover the sets. The interval or would also contain the point 0, and so the length of zero should also be less than 1/2 and less than 1/4.
The length, or measure, of a set is defined to be the smallest interval length (or sum of lengths, if we use more than one interval) that contains the set we care about.^{4}
Going back to the single point example, the point 0 is contained in , but also in or . In other words, we can cover it with intervals of smaller and smaller lengths, heading towards zero. Thus, by our definition, the measure of the set , the single point 0, is zero. In other words, a single point has no length.
The same kind of argument works for any finite set of points. You take smaller and smaller intervals around each of the points, and so the total length of the intervals go to zero.
This shows that the measure of any finite set is zero.
What happens if we move to infinite set?
The simplest infinite set is the counting numbers, 1, 2, 3, etc.. If we try the exact same thing we did with a finite number of points and cover each point with an interval of the same length, no matter how small each individual interval is, the total length of intervals would be infinite.^{5}
This is not wrong, per se, but remember that our sum of interval lengths is supposed to be an estimate of the length of our set. It’s possible the counting numbers should have infinite length, but let’s see if we can do better than that.
To try to improve our estimate, we’ll put smaller and smaller intervals around each subsequent number. So, we cover 1 with (length 1), cover 2 with (length 1/2), cover 3 with (length 1/4), etc.
Thus, we managed to cover all the counting numbers with a collection of intervals of total length .^{6} That means the measure (i.e., length) of the counting numbers is less than 2.
Of course, we didn’t have to start with an interval of width 1. We could have covered 1 with , 2 with , etc. In this case, we’d have a total length of . So the measure of the counting numbers is less than 1.
Continuing this idea, we could start with smaller and smaller intervals, and end up with a total length of 1/2 or 1/4 or 1/8 and so on. By our definition, the length of the counting numbers has to be zero!
In fact, this same idea works for any countable set, i.e., any set you can count. You cover the first point with a small interval, the second point with a smaller interval, etc. until you’ve covered them all. Then, you try again with even smaller intervals. Thus, the measure of any countable set is zero!
Now, this doesn’t seem very interesting when worded like this, but let me give you a slightly more amazing example.
A rational number is a number that can be written as the fraction of two counting numbers, maybe with a minus sign. So, 3/4 and -12374/421 are rational numbers, but is not. Since any number can be approximated by a rational number (e.g. ), the rational numbers are everywhere.
There are so many rational numbers that, no matter which two numbers you pick, there are infinitely many rational numbers between those two numbers.
It’s surprising, then, that there are only as many rational numbers as there are counting numbers! More details are available in The size of infinity, but the basic idea is that you can line up the rational numbers so that you can count them. In other words, you can list them in a definite order — a first rational number, a second, a third, etc.
But since we can order them in this way, we can put an interval of length 1 on the first rational number, an interval of length 1/2 on the second, an interval of length 1/4 on the third, etc. Thus, the length of all the rational numbers is no more than 2.
And, of course, we can use smaller and smaller intervals, and thus show that the rational numbers have no length at all!
Bizarre, right? Numbers can be dense (which is the technical way to say they’re everywhere, no matter how far you zoom in), but still be so close to nothing that they have no length at all!
The length of (countable) infinity is always zero!
What about larger infinities? We’ll talk about that more next time.
- The proof is in the post A bigger infinity. It’s really cool that there is more than one size of infinity. In fact, there are infinitely many different sizes of infinity. For each size of infinity (like the size of the real numbers), there is a larger infinity. We talked more about that in the post An even biggerer infinity. ↩
- Intervals containing their endpoints are called closed, while those not containing their endpoints are called open. ↩
- Well, technically not any set of points — there are unmeasureable sets, but they’re weird and unusual. In fact, you can’t even prove they exist using some standard sets of axioms. I’ll probably talk more about these sets in a later post. ↩
- In other words, the infimum of the sum of interval lengths, where the infimum is taken over all (countable) collections of intervals that cover the set. ↩
- Importantly, you cannot just think of each point as having length zero, and then adding up infinitely many zeros to get zero. That… doesn’t make sense. What you’re trying to say is that the length should be . But anything times zero is zero, and anything times infinity is infinity. So what should be? Zero or infinity? Or something else? The fact of the matter is that we need more information. Sometimes it makes sense to say it is zero, sometimes infinity, and sometimes some other number, like 7. We need to carefully use our definition of length in order to know which one is correct for this circumstance. ↩
- In case you’ve never seen this before, suppose that added up to some number . Well, then if we multiply that number by 2, that’s the same as multiplying all of those numbers together and then adding them up. Thus . But, , so if we subtract the two sums, the 1’s cancel out, the 1/2’s cancel out, the 1/4’s cancel out, etc., and we’re left with . Thus this sum is 2. ↩