How Big Is Infinity?
-
0:14 - 0:17When I was in fourth grade,
my teacher said to us one day: -
0:17 - 0:19"There are as many even numbers
as there are numbers." -
0:20 - 0:21"Really?", I thought.
-
0:21 - 0:23Well, yeah, there are
infinitely many of both, -
0:23 - 0:26so I suppose there are
the same number of them. -
0:26 - 0:29But even numbers are only part
of the whole numbers, -
0:29 - 0:31all the odd numbers are left over,
-
0:31 - 0:34so there's got to be more whole numbers
than even numbers, right? -
0:34 - 0:36To see what my teacher was getting at,
-
0:36 - 0:39let's first think about what it means
for two sets to be the same size. -
0:39 - 0:42What do I mean when I say
I have the same number of fingers -
0:42 - 0:44on my right hand as I do on left hand?
-
0:44 - 0:48Of course, I have five fingers on each,
but it's actually simpler than that. -
0:48 - 0:52I don't have to count, I only need to see
that I can match them up, one to one. -
0:53 - 0:55In fact, we think that some ancient people
-
0:55 - 0:58who spoke languages that didn't have words
for numbers greater than three -
0:58 - 0:59used this sort of magic.
-
1:00 - 1:02For instance, if you let
your sheep out of a pen to graze, -
1:02 - 1:06you can keep track of how many went out
by setting aside a stone for each one, -
1:06 - 1:09and putting those stones back
one by one when the sheep return, -
1:09 - 1:12so you know if any are missing
without really counting. -
1:12 - 1:15As another example of matching being
more fundamental than counting, -
1:15 - 1:17if I'm speaking to a packed auditorium,
-
1:17 - 1:20where every seat is taken
and no one is standing, -
1:20 - 1:23I know that there are the same number
of chairs as people in the audience, -
1:23 - 1:26even though I don't know
how many there are of either. -
1:26 - 1:29So, what we really mean when we say
that two sets are the same size -
1:29 - 1:31is that the elements in those sets
-
1:31 - 1:33can be matched up one by one in some way.
-
1:33 - 1:35My fourth grade teacher showed us
-
1:35 - 1:38the whole numbers laid out in a row,
and below each we have its double. -
1:38 - 1:41As you can see, the bottom row
contains all the even numbers, -
1:41 - 1:42and we have a one-to-one match.
-
1:42 - 1:45That is, there are as many
even numbers as there are numbers. -
1:45 - 1:48But what still bothers us is our distress
-
1:48 - 1:51over the fact that even numbers
seem to be only part of the whole numbers. -
1:51 - 1:53But does this convince you
-
1:53 - 1:55that I don't have
the same number of fingers -
1:55 - 1:57on my right hand as I do on my left?
-
1:57 - 1:58Of course not.
-
1:58 - 2:00It doesn't matter if you try to match
-
2:00 - 2:02the elements in some way
and it doesn't work, -
2:02 - 2:04that doesn't convince us of anything.
-
2:04 - 2:05If you can find one way
-
2:05 - 2:07in which the elements
of two sets do match up, -
2:07 - 2:10then we say those two sets have
the same number of elements. -
2:10 - 2:12Can you make a list of all the fractions?
-
2:13 - 2:15This might be hard,
there are a lot of fractions! -
2:15 - 2:17And it's not obvious what to put first,
-
2:17 - 2:19or how to be sure
all of them are on the list. -
2:19 - 2:22Nevertheless, there is a very clever way
-
2:22 - 2:24that we can make a list
of all the fractions. -
2:24 - 2:28This was first done by Georg Cantor,
in the late eighteen hundreds. -
2:28 - 2:31First, we put all
the fractions into a grid. -
2:31 - 2:32They're all there.
-
2:32 - 2:36For instance, you can find, say, 117/243,
-
2:36 - 2:39in the 117th row and 243rd column.
-
2:39 - 2:41Now we make a list out of this
-
2:41 - 2:44by starting at the upper left
and sweeping back and forth diagonally, -
2:44 - 2:47skipping over any fraction, like 2/2,
-
2:47 - 2:50that represents the same number
as one the we've already picked. -
2:50 - 2:52We get a list of all the fractions,
-
2:52 - 2:54which means we've created
a one-to-one match -
2:54 - 2:56between the whole numbers
and the fractions, -
2:56 - 2:59despite the fact that we thought
maybe there ought to be more fractions. -
2:59 - 3:01OK, here's where it gets
really interesting. -
3:01 - 3:03You may know that not all real numbers
-
3:03 - 3:06-- that is, not all the numbers
on a number line -- are fractions. -
3:07 - 3:09The square root
of two and pi, for instance. -
3:09 - 3:11Any number like this is called irrational.
-
3:11 - 3:13Not because it's crazy, or anything,
-
3:13 - 3:16but because the fractions are
ratios of whole numbers, -
3:16 - 3:18and so are called rationals;
-
3:18 - 3:21meaning the rest are
non-rational, that is, irrational. -
3:21 - 3:25Irrationals are represented
by infinite, non-repeating decimals. -
3:25 - 3:27So, can we make a one-to-one match
-
3:27 - 3:30between the whole numbers
and the set of all the decimals, -
3:30 - 3:32both the rationals and the irrationals?
-
3:32 - 3:34That is, can we make
a list of all the decimal numbers? -
3:34 - 3:36Cantor showed that you can't.
-
3:36 - 3:40Not merely that we don't know how,
but that it can't be done. -
3:40 - 3:44Look, suppose you claim you have made
a list of all the decimals. -
3:44 - 3:46I'm going to show you
that you didn't succeed, -
3:46 - 3:48by producing a decimal
that is not on your list. -
3:48 - 3:51I'll construct my decimal
one place at a time. -
3:51 - 3:53For the first decimal place of my number,
-
3:53 - 3:56I'll look at the first decimal place
of your first number. -
3:56 - 3:58If it's a one, I'll make mine a two;
-
3:58 - 4:00otherwise I'll make mine a one.
-
4:00 - 4:03For the second place of my number,
-
4:03 - 4:05I'll look at the second place
of your second number. -
4:05 - 4:08Again, if yours is a one,
I'll make mine a two, -
4:08 - 4:10and otherwise I'll make mine a one.
-
4:10 - 4:11See how this is going?
-
4:11 - 4:14The decimal I've produced
can't be on your list. -
4:14 - 4:18Why? Could it be, say, your 143rd number?
-
4:18 - 4:21No, because the 143rd place of my decimal
-
4:21 - 4:24is different from the 143rd place
of your 143rd number. -
4:24 - 4:26I made it that way.
-
4:26 - 4:27Your list is incomplete.
-
4:27 - 4:29It doesn't contain my decimal number.
-
4:30 - 4:32And, no matter what list you give me,
I can do the same thing, -
4:32 - 4:35and produce a decimal
that's not on that list. -
4:35 - 4:37So we're faced with this
astounding conclusion: -
4:37 - 4:40The decimal numbers
cannot be put on a list. -
4:40 - 4:44They represent a bigger infinity
that the infinity of whole numbers. -
4:44 - 4:47So, even though we're familiar
with only a few irrationals, -
4:47 - 4:49like square root of two and pi,
-
4:49 - 4:50the infinity of irrationals
-
4:50 - 4:53is actually greater
than the infinity of fractions. -
4:53 - 4:54Someone once said that the rationals
-
4:54 - 4:57-- the fractions -- are
like the stars in the night sky. -
4:58 - 5:01The irrationals are like the blackness.
-
5:01 - 5:04Cantor also showed that,
for any infinite set, -
5:04 - 5:07forming a new set made of
all the subsets of the original set -
5:07 - 5:10represents a bigger infinity
than that original set. -
5:10 - 5:12This means that,
once you have one infinity, -
5:12 - 5:14you can always make a bigger one
-
5:14 - 5:17by making the set of all subsets
of that first set. -
5:17 - 5:18And then an even bigger one
-
5:18 - 5:21by making the set
of all the subsets of that one. -
5:21 - 5:22And so on.
-
5:22 - 5:26And so, there are an infinite number
of infinities of different sizes. -
5:26 - 5:29If these ideas make you
uncomfortable, you are not alone. -
5:29 - 5:32Some of the greatest
mathematicians of Cantor's day -
5:32 - 5:33were very upset with this stuff.
-
5:33 - 5:36They tried to make these different
infinities irrelevant, -
5:36 - 5:38to make mathematics work
without them somehow. -
5:38 - 5:40Cantor was even vilified personally,
-
5:40 - 5:43and it got so bad for him
that he suffered severe depression, -
5:43 - 5:46and spent the last half of his life
in and out of mental institutions. -
5:46 - 5:49But eventually, his ideas won out.
-
5:49 - 5:52Today, they're considered
fundamental and magnificent. -
5:52 - 5:54All research mathematicians
accept these ideas, -
5:54 - 5:56every college math major learns them,
-
5:56 - 5:58and I've explained them
to you in a few minutes. -
5:58 - 6:01Some day, perhaps,
they'll be common knowledge. -
6:01 - 6:02There's more.
-
6:02 - 6:04We just pointed out
that the set of decimal numbers -
6:05 - 6:07-- that is, the real numbers --
is a bigger infinity -
6:07 - 6:08than the set of whole numbers.
-
6:08 - 6:11Cantor wondered
whether there are infinities -
6:11 - 6:13of different sizes
between these two infinities. -
6:13 - 6:15He didn't believe there were,
but couldn't prove it. -
6:15 - 6:18Cantor's conjecture became known
as the continuum hypothesis. -
6:19 - 6:22In 1900, the great
mathematician David Hilbert -
6:22 - 6:24listed the continuum hypothesis
-
6:24 - 6:26as the most important
unsolved problem in mathematics. -
6:26 - 6:29The 20th century saw
a resolution of this problem, -
6:29 - 6:32but in a completely unexpected,
paradigm-shattering way. -
6:33 - 6:35In the 1920s, Kurt Gödel showed
-
6:35 - 6:38that you can never prove
that the continuum hypothesis is false. -
6:38 - 6:41Then, in the 1960s, Paul J. Cohen showed
-
6:41 - 6:44that you can never prove
that the continuum hypothesis is true. -
6:44 - 6:46Taken together, these results mean
-
6:46 - 6:49that there are unanswerable
questions in mathematics. -
6:49 - 6:50A very stunning conclusion.
-
6:50 - 6:53Mathematics is rightly considered
the pinnacle of human reasoning, -
6:53 - 6:57but we now know that even mathematics
has its limitations. -
6:57 - 7:01Still, mathematics has some truly
amazing things for us to think about.
- Title:
- How Big Is Infinity?
- Speaker:
- Dennis Wildfogel
- Description:
-
View full lesson: http://ed.ted.com/lessons/how-big-is-infinity
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.
Lesson by Dennis Wildfogel, animation by Augenblick Studios.
- Video Language:
- English
- Team:
- closed TED
- Project:
- TED-Ed
- Duration:
- 07:13
Michelle Mehrtens edited English subtitles for How big is infinity? | ||
Maria Ruzsane Cseresnyes commented on English subtitles for How big is infinity? | ||
Krystian Aparta edited English subtitles for How big is infinity? | ||
Krystian Aparta commented on English subtitles for How big is infinity? | ||
grankabeza added a translation |
Krystian Aparta
The English transcript was updated on 2/13/2015.
Maria Ruzsane Cseresnyes
"between the whole numbers
and the set of all the decimals" To the construction they sould suppose, that 0 <= the all the decimals <= 1.