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When I was in fourth grade, my teacher said to us one day:
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"There are as many even numbers as there are numbers."
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"Really?", I thought. Well, yeah, there are infinitely many of both, so I suppose there are the same number of them.
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But, on the other hand, even numbers are only part of the whole numbers, all the odd numbers are left over,
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so there's got to be more whole numbers than even numbers, right?
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To see what my teacher was getting at, let's first think about what it means for two sets to be the same size.
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What do I mean when I say I have the same number of fingers on my right hand as I do on left hand?
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Of course, I have five fingers on each, but it's actually simpler than that.
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I don't have to count, I only need to see that I can match them up, one to one.
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In fact, we think that some ancient people who spoke languages that didn't have words
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for numbers greater than three used this sort of magic. For instance, if you let your sheep out of a pen to graze,
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you can keep track of how many went out by setting aside a stone for each one,
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and then putting those stones back one by one when the sheep return,
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so you know if any are missing without really counting.
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As another example of matching being more fundamental than counting,
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if I'm speaking to a packed auditorium, where every seat is taken and no one is standing,
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I know that there are the same number of chairs as people in the audience,
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even though I don't know how many there are of either.
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So, what we really mean when we say that two sets are the same size
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is that the elements in those sets can be matched up one by one in some way.
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So my fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double.
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As you can see, the bottom row contains all the even numbers, and we have a one-to-one match.
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That is, there are as many even numbers as there are numbers.
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But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers.
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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?
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Of course not. It doesn't matter if you try to match the elements in some way and that doesn't work,
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that doesn't convince us of anything.
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If you can find one way in which the elements of two sets do match up,
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then we say those two sets have the same number of elements.
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Can you make a list of all the fractions? This might be hard, there are a lot of fractions!
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And it's not obvious what to put first, or how to be sure all of them are on the list.
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Nevertheless, there is a very clever way that we can make a list of all the fractions.
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This was first done by Georg Cantor, in the late eighteen hundreds.
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First, we put all the fractions into a grid. They're all there. For instance, you can find, say, 117/243,
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in the 117th row and 223rd column.
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Now we make a list out of this by starting at the upper left and sweeping back and forth diagonally,
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skipping over any fraction, like 2/2, that represents the same number as one the we've already picked.
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And so we get a list of all the fractions, which means we've created a one-to-one match
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between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions.
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Ok, here's where it gets really interesting.
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You may know that not all real numbers —that is, not all the numbers on a number line— are fractions.
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The square root of two and pi, for instance.
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Any number like this is called irrational. Not because it's crazy, or anything, but because the fractions are
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ratios of whole numbers, and so are called rationals; meaning the rest are non-rational, that is, irrational.
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Irrationals are represented by infinite, non-repeating decimals.
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So, can we make a one-to-one match between the whole numbers and the set of all the decimals,
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both the rationals and the irrationals? That is, can we make a list of all the decimal numbers?
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Candor showed that you can't. Not merely that we don't know how, but that it can't be done.
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Look, suppose you claim you have made a list of all the decimals. I'm going to show you that you didn't succeed,
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by producing a decimal that is not on your list.
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