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• When I was in 4th 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 on the other hand, the 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 my left hand?

• Of course, I've 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 matching.

• 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 then putting those stones back one by one when the sheep return,

• so that 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.

• So my 4th 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 the 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 over 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 we've already picked.

• And so 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 numbersthat is, not all the numbers on a number lineare 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's 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 1, I'll make mine a 2.

• Otherwise, I'll make mine a 1.

• For the second place of my number,

• I'll look at the second place of your second number.

• Again, if yours is a 1, I'll make mine a 2,

• and otherwise i'll make mine a 1.

• See how this is going?

• The decimal I produce 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 than 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 rationalsthe fractionsare 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 a set of all subsets of that first set.

• And then an even bigger one

• by making a set of all 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're 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.

• He spent the last half of his life in and out of mental institutions.

• But eventually, his ideas won out.

• Today they are 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.

• Someday, perhaps, they'll be common knowledge.

• There's more.

• We just pointed out that the set of decimal numbers

• that is, the real numbersis a bigger infinity than the set of whole numbers.

• Cantor wondered if 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 Godel 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 had its limitations.

• Still, mathematics has some truly amazing things for us to think about.

When I was in 4th grade, my teacher said to us one day:

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# 【TED-Ed】How Big Is Infinity? - Dennis Wildfogel

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Furong Lai posted on 2014/01/11
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