Using the fundamentals of set theory, explore the mind-bending concept of the “infinity of infinities” — and how it led mathematicians to conclude that math itself contains unanswerable questions.
When I was in fourth grade, my teacher said to us one day: “There are as many even numbers as there are numbers.”
“Really?”, I thought.
Well, yeah, there are infinitely many of both, so I suppose there are the same number of them.
But even numbers are only part of the whole numbers, all the odd numbers are left over, so there’s got to be more whole numbers than even numbers, right?
To see what my teacher was getting at, let’s first think about what it means for two sets to be the same size.
What do I mean when I say I have the same number of fingers on my right hand as I do on left hand? Of course, I have five fingers on each, but it’s actually simpler than that. I don’t have to count, I only need to see that I can match them up, one to one.
In fact, we think that some ancient people who spoke languages that didn’t have words for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and putting those stones back one by one when the sheep return, so you know if any are missing without really counting.
As another example of matching being more fundamental than counting, if I’m speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don’t know how many there are of either.
So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way. My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double.
As you can see, the bottom row contains all the even numbers, and we have a one-to-one match. That is, there are as many even numbers as there are numbers.
But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers. But does this convince you that I don’t have the same number of fingers on my right hand as I do on my left? Of course not.
It doesn’t matter if you try to match the elements in some way and it doesn’t work, that doesn’t convince us of anything. If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements.
Can you make a list of all the fractions?
This might be hard, there are a lot of fractions! And it’s not obvious what to put first, or how to be sure all of them are on the list. Nevertheless, there is a very clever way that we can make a list of all the fractions. This was first done by Georg Cantor, in the late 1800s.
First, we put all the fractions into a grid. They’re all there. For instance, you can find, say, 117/243, in the 117th row and 243rd column.
Now we make a list out of this by starting at the upper left and sweeping back and forth diagonally, skipping over any fraction, like 2/2, that represents the same number as one the we’ve already picked.
We get a list of all the fractions, which means we’ve created a one-to-one match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions.
OK, here’s where it gets really interesting. You may know that not all real numbers — that is, not all the numbers on a number line — are fractions. The square root of two and pi, for instance. Any number like this is called irrational. Not because it’s crazy, or anything, but because the fractions are ratios of whole numbers, and so are called rationals; meaning the rest are non-rational, that is, irrational.
Irrationals are represented by infinite, non-repeating decimals. So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the irrationals? That is, can we make a list of all the decimal numbers? Cantor showed that you can’t.
Not merely that we don’t know how, but that it can’t be done. Look, suppose you claim you have made a list of all the decimals. I’m going to show you that you didn’t succeed, by producing a decimal that is not on your list. I’ll construct my decimal one place at a time.
For the first decimal place of my number, I’ll look at the first decimal place of your first number. If it’s a one, I’ll make mine a two; otherwise I’ll make mine a one.
For the second place of my number, I’ll look at the second place of your second number. Again, if yours is a one, I’ll make mine a two, and otherwise I’ll make mine a one.
See how this is going? The decimal I’ve produced can’t be on your list. Why? Could it be, say, your 143rd number? No, because the 143rd place of my decimal is different from the 143rd place of your 143rd number. I made it that way.
Your list is incomplete. It doesn’t contain my decimal number. And, no matter what list you give me, I can do the same thing, and produce a decimal that’s not on that list.
So we’re faced with this astounding conclusion: The decimal numbers cannot be put on a list. They represent a bigger infinity that the infinity of whole numbers.
So, even though we’re familiar with only a few irrationals, like square root of two and pi, the infinity of irrationals is actually greater than the infinity of fractions.
Someone once said that the rationals — the fractions — are like the stars in the night sky. The irrationals are like the blackness. Cantor also showed that, for any infinite set, forming a new set made of all the subsets of the original set represents a bigger infinity than that original set.
This means that, once you have one infinity, you can always make a bigger one by making the set of all subsets of that first set. And then an even bigger one by making the set of all the subsets of that one. And so on.
And so, there are an infinite number of infinities of different sizes. If these ideas make you uncomfortable, you are not alone. Some of the greatest mathematicians of Cantor’s day were very upset with this stuff. They tried to make these different infinities irrelevant, to make mathematics work without them somehow.
Cantor was even vilified personally, and it got so bad for him that he suffered severe depression, and spent the last half of his life in and out of mental institutions. But eventually, his ideas won out.
Today, they’re considered fundamental and magnificent. All research mathematicians accept these ideas, every college math major learns them, and I’ve explained them to you in a few minutes.
Some day, perhaps, they’ll be common knowledge. There’s more. We just pointed out that the set of decimal numbers — that is, the real numbers — is a bigger infinity than the set of whole numbers.
Cantor wondered whether there are infinities of different sizes between these two infinities. He didn’t believe there were, but couldn’t prove it. Cantor’s conjecture became known as the continuum hypothesis.
In 1900, the great mathematician David Hilbert listed the continuum hypothesis as the most important unsolved problem in mathematics. The 20th century saw a resolution of this problem, but in a completely unexpected, paradigm-shattering way.
In the 1920s, Kurt Gödel showed that you can never prove that the continuum hypothesis is false. Then, in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true.
Taken together, these results mean that there are unanswerable questions in mathematics. A very stunning conclusion.
Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations. Still, mathematics has some truly amazing things for us to think about.
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