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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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B1 INT US decimal number list cantor infinity mathematics

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

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    Furong Lai   posted on 2014/01/11
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