[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:17.50,0:00:19.77,Default,,0000,0000,0000,,Numbers are strange.
Dialogue: 0,0:00:20.67,0:00:22.92,Default,,0000,0000,0000,,They are not physical objects.
Dialogue: 0,0:00:22.92,0:00:28.54,Default,,0000,0000,0000,,No one has bumped into the number two\Nor tripped over the number three;
Dialogue: 0,0:00:28.54,0:00:30.100,Default,,0000,0000,0000,,not even your crazy math professor.
Dialogue: 0,0:00:32.45,0:00:35.97,Default,,0000,0000,0000,,They are not mental objects either.
Dialogue: 0,0:00:35.97,0:00:39.23,Default,,0000,0000,0000,,The thought of your beloved \Nisn't your beloved
Dialogue: 0,0:00:39.23,0:00:41.81,Default,,0000,0000,0000,,no matter how much\Nyou might want it to be.
Dialogue: 0,0:00:41.81,0:00:46.38,Default,,0000,0000,0000,,And no more is the thought \Nof the number three, the number three.
Dialogue: 0,0:00:47.09,0:00:50.72,Default,,0000,0000,0000,,Nor do numbers exist in space or time.
Dialogue: 0,0:00:50.72,0:00:55.03,Default,,0000,0000,0000,,You don't expect to find the number three\Nin the kitchen cupboard,
Dialogue: 0,0:00:55.03,0:00:57.34,Default,,0000,0000,0000,,and you don't need to worry
Dialogue: 0,0:00:57.34,0:01:02.53,Default,,0000,0000,0000,,that numbers once didn't exist\Nor might one day cease to exist.
Dialogue: 0,0:01:03.26,0:01:06.54,Default,,0000,0000,0000,,But even though numbers \Nare far removed\N
Dialogue: 0,0:01:06.54,0:01:09.80,Default,,0000,0000,0000,,from the familiar world\Nof thoughts and things,\N
Dialogue: 0,0:01:10.89,0:01:17.57,Default,,0000,0000,0000,,they're intimately connected to that world\Nbecause we do things with numbers.
Dialogue: 0,0:01:18.30,0:01:22.42,Default,,0000,0000,0000,,We count with them, we measure with them,
Dialogue: 0,0:01:22.42,0:01:26.12,Default,,0000,0000,0000,,we formulate\Nour scientific theories with them.
Dialogue: 0,0:01:27.48,0:01:31.21,Default,,0000,0000,0000,,So this makes it \Nall the stranger what they are.
Dialogue: 0,0:01:31.21,0:01:35.90,Default,,0000,0000,0000,,How can they be so far removed \Nfrom the familiar world
Dialogue: 0,0:01:35.90,0:01:39.38,Default,,0000,0000,0000,,and yet so intimately connected to it?
Dialogue: 0,0:01:40.40,0:01:45.95,Default,,0000,0000,0000,,In this talk I want to consider \Nthree views about the nature of number
Dialogue: 0,0:01:45.95,0:01:49.88,Default,,0000,0000,0000,,that were developed \Nby mathematicians and philosophers
Dialogue: 0,0:01:49.88,0:01:55.90,Default,,0000,0000,0000,,around the end of the 19th century\Nand the beginning of the 20th century.
Dialogue: 0,0:01:56.68,0:02:00.49,Default,,0000,0000,0000,,All of these views presuppose \Nthat strictly speaking\N
Dialogue: 0,0:02:00.49,0:02:05.49,Default,,0000,0000,0000,,what we count are not things,\Nbut sets of things.
Dialogue: 0,0:02:05.49,0:02:11.53,Default,,0000,0000,0000,,A set is just many things, any things\Nyou like, considered as one.
Dialogue: 0,0:02:11.53,0:02:18.04,Default,,0000,0000,0000,,So for example, we have the set \Nof beer bottles that you drank last night.
Dialogue: 0,0:02:19.68,0:02:22.94,Default,,0000,0000,0000,,The bottles are put in these braces \Nto indicate that the six bottles
Dialogue: 0,0:02:22.94,0:02:25.67,Default,,0000,0000,0000,,are being considered as one object.
Dialogue: 0,0:02:25.67,0:02:31.65,Default,,0000,0000,0000,,Then we have the set consisting \Nof your two favorite pets, Fido and Felix.
Dialogue: 0,0:02:33.40,0:02:38.20,Default,,0000,0000,0000,,Or we have a set consisting \Nof all the natural numbers,
Dialogue: 0,0:02:38.20,0:02:40.42,Default,,0000,0000,0000,,so they're put together\Nin this very big set:
Dialogue: 0,0:02:40.42,0:02:42.72,Default,,0000,0000,0000,,0, 1, 2, 3, 4 and so on.
Dialogue: 0,0:02:44.23,0:02:49.00,Default,,0000,0000,0000,,And so what we do when we count \Nis associate a number with a set.
Dialogue: 0,0:02:49.00,0:02:52.26,Default,,0000,0000,0000,,In the case of the beer bottles, \Nthe number six,
Dialogue: 0,0:02:52.26,0:02:55.78,Default,,0000,0000,0000,,assuming you're not too drunk\Nto count them.
Dialogue: 0,0:02:56.84,0:02:59.56,Default,,0000,0000,0000,,In the case of your pets, the number two.
Dialogue: 0,0:03:00.04,0:03:04.67,Default,,0000,0000,0000,,And in the case of the natural numbers,\Nwhen we put them into one big set,
Dialogue: 0,0:03:04.67,0:03:06.81,Default,,0000,0000,0000,,is going to be some infinite number.
Dialogue: 0,0:03:07.87,0:03:12.10,Default,,0000,0000,0000,,The first view I want to consider\Nabout the nature of numbers
Dialogue: 0,0:03:12.10,0:03:16.68,Default,,0000,0000,0000,,was developed independently \Nby two great philosopher mathematicians,
Dialogue: 0,0:03:16.68,0:03:19.66,Default,,0000,0000,0000,,Gottlob Frege and Bertrand Russell.
Dialogue: 0,0:03:19.66,0:03:22.95,Default,,0000,0000,0000,,These two individuals\Nwere very different from one another.
Dialogue: 0,0:03:23.72,0:03:27.08,Default,,0000,0000,0000,,Russell came from the English aristocracy;\N
Dialogue: 0,0:03:27.08,0:03:30.12,Default,,0000,0000,0000,,Frege from the comfortable\NGerman middle class.
Dialogue: 0,0:03:30.89,0:03:34.21,Default,,0000,0000,0000,,Russell was a crusading liberal;
Dialogue: 0,0:03:35.23,0:03:38.36,Default,,0000,0000,0000,,Frege, I'm sorry to say, was a proto-Nazi.
Dialogue: 0,0:03:39.33,0:03:45.48,Default,,0000,0000,0000,,Russell had four wives, \Nand innumerable mistresses;
Dialogue: 0,0:03:45.48,0:03:47.30,Default,,0000,0000,0000,,Frege had a single wife, \N
Dialogue: 0,0:03:47.30,0:03:52.53,Default,,0000,0000,0000,,and as far as I know, \Nenjoyed a happy, staid marital existence.
Dialogue: 0,0:03:53.16,0:03:55.61,Default,,0000,0000,0000,,But despite these differences,\N
Dialogue: 0,0:03:55.61,0:03:58.68,Default,,0000,0000,0000,,they had more or less the same view\Nabout the nature of number.
Dialogue: 0,0:03:59.39,0:04:00.71,Default,,0000,0000,0000,,So what was it?
Dialogue: 0,0:04:00.71,0:04:04.09,Default,,0000,0000,0000,,Well let's take the number two\Nas an example.
Dialogue: 0,0:04:04.09,0:04:09.12,Default,,0000,0000,0000,,Two can be used to number\Nany two-membered set or pair.
Dialogue: 0,0:04:09.12,0:04:15.55,Default,,0000,0000,0000,,So it can be used to number the set\Nwhose members are Frege and Russell.
Dialogue: 0,0:04:15.55,0:04:18.04,Default,,0000,0000,0000,,Or it can be used to number the set
Dialogue: 0,0:04:18.04,0:04:22.22,Default,,0000,0000,0000,,consisting of your favorite pets, \NFido and Felix.
Dialogue: 0,0:04:23.18,0:04:25.50,Default,,0000,0000,0000,,Or it can be used to number
Dialogue: 0,0:04:25.50,0:04:29.25,Default,,0000,0000,0000,,Dickens' famous two cities,\NLondon and Paris.
Dialogue: 0,0:04:29.25,0:04:32.35,Default,,0000,0000,0000,,I insisted that London \Nbe placed first there.
Dialogue: 0,0:04:32.35,0:04:33.64,Default,,0000,0000,0000,,(Laughter)
Dialogue: 0,0:04:35.48,0:04:42.23,Default,,0000,0000,0000,,Now the idea of Russell and Frege was to\Nput all of these pairs into one big set.
Dialogue: 0,0:04:42.93,0:04:47.22,Default,,0000,0000,0000,,We pile them all into one big set, \Nand that would be the number two.
Dialogue: 0,0:04:47.22,0:04:50.23,Default,,0000,0000,0000,,So the number two \Nwould be a set of sets,
Dialogue: 0,0:04:50.23,0:04:55.36,Default,,0000,0000,0000,,and these sets would just be all the pairs\Nthat could be counted by the number two.
Dialogue: 0,0:04:55.36,0:04:57.65,Default,,0000,0000,0000,,Similarly, for all other numbers,
Dialogue: 0,0:04:57.65,0:05:00.10,Default,,0000,0000,0000,,the number three \Nwould be the set of all triples,
Dialogue: 0,0:05:00.10,0:05:04.42,Default,,0000,0000,0000,,the number four the set \Nof all quadruples, and so on.
Dialogue: 0,0:05:04.42,0:05:07.17,Default,,0000,0000,0000,,A simple and beautiful theory.
Dialogue: 0,0:05:08.03,0:05:11.74,Default,,0000,0000,0000,,Unfortunately, it led to contradiction.
Dialogue: 0,0:05:12.59,0:05:15.53,Default,,0000,0000,0000,,I can't give a demonstration \Nof the contradiction here,
Dialogue: 0,0:05:15.53,0:05:18.74,Default,,0000,0000,0000,,but I can give you a feel\Nfor how it arose.
Dialogue: 0,0:05:19.79,0:05:24.83,Default,,0000,0000,0000,,You'll recall that the number two\Nwas the set of all pairs,
Dialogue: 0,0:05:24.83,0:05:27.06,Default,,0000,0000,0000,,all pairs of whatever.
Dialogue: 0,0:05:27.06,0:05:33.14,Default,,0000,0000,0000,,So in particular, it would include pairs\Nthat themselves contain the number two.
Dialogue: 0,0:05:33.91,0:05:35.94,Default,,0000,0000,0000,,So let's look at the particular such pair,
Dialogue: 0,0:05:35.94,0:05:39.20,Default,,0000,0000,0000,,the pair consisting of the number two\Nand the number one.
Dialogue: 0,0:05:41.67,0:05:45.85,Default,,0000,0000,0000,,Then that pair, the pair {1, 2},\Nwould itself be inside the number two.
Dialogue: 0,0:05:45.85,0:05:49.38,Default,,0000,0000,0000,,So the number two would contain itself, \N
Dialogue: 0,0:05:50.03,0:05:52.54,Default,,0000,0000,0000,,and that looks as if it's impossible.
Dialogue: 0,0:05:53.31,0:05:55.29,Default,,0000,0000,0000,,So here's an analogy:
Dialogue: 0,0:05:55.29,0:06:00.29,Default,,0000,0000,0000,,imagine a very hungry serpent \Nthat tries to eat its own tail.
Dialogue: 0,0:06:00.29,0:06:03.29,Default,,0000,0000,0000,,Now, it could succeed in doing this
Dialogue: 0,0:06:03.29,0:06:06.74,Default,,0000,0000,0000,,- this is the best we could do\Nby way of illustration -
Dialogue: 0,0:06:06.74,0:06:09.32,Default,,0000,0000,0000,,This is gross, but still possible.
Dialogue: 0,0:06:09.32,0:06:10.77,Default,,0000,0000,0000,,(Laughter)
Dialogue: 0,0:06:10.77,0:06:14.20,Default,,0000,0000,0000,,But imagine now\Nthat the serpent is so ravenous
Dialogue: 0,0:06:14.20,0:06:18.02,Default,,0000,0000,0000,,that it attempts to eat itself\Nin its entirety.
Dialogue: 0,0:06:18.74,0:06:20.68,Default,,0000,0000,0000,,That's not even possible
Dialogue: 0,0:06:20.68,0:06:26.01,Default,,0000,0000,0000,,because then, the serpent's stomach\Nwould have to be inside its stomach.
Dialogue: 0,0:06:26.01,0:06:29.04,Default,,0000,0000,0000,,And that's what happens \Nwith the number two.
Dialogue: 0,0:06:29.04,0:06:35.14,Default,,0000,0000,0000,,The number two, as you see,\Nhas itself inside of its very own stomach.
Dialogue: 0,0:06:36.29,0:06:38.14,Default,,0000,0000,0000,,What was to be done?
Dialogue: 0,0:06:39.10,0:06:44.01,Default,,0000,0000,0000,,The mathematician John von Neumann\Ncame up with a brilliant solution;
Dialogue: 0,0:06:44.59,0:06:48.47,Default,,0000,0000,0000,,von Neumann was perhaps \None of the most versatile mathematicians\N
Dialogue: 0,0:06:48.47,0:06:50.16,Default,,0000,0000,0000,,who ever existed.
Dialogue: 0,0:06:50.16,0:06:54.32,Default,,0000,0000,0000,,He helped invent game theory\Nand the modern computer.
Dialogue: 0,0:06:55.23,0:06:58.46,Default,,0000,0000,0000,,He was a prodigy
Dialogue: 0,0:06:58.46,0:07:01.46,Default,,0000,0000,0000,,and had the most amazing\Ncomputational skills.
Dialogue: 0,0:07:01.46,0:07:04.58,Default,,0000,0000,0000,,So what was his solution?
Dialogue: 0,0:07:04.58,0:07:05.66,Default,,0000,0000,0000,,There he is.
Dialogue: 0,0:07:05.66,0:07:07.21,Default,,0000,0000,0000,,He said: "Well look,
Dialogue: 0,0:07:07.21,0:07:13.05,Default,,0000,0000,0000,,rather than take the number two\Nto be the set of all pairs,
Dialogue: 0,0:07:13.05,0:07:16.48,Default,,0000,0000,0000,,take it to be a particular pair."
Dialogue: 0,0:07:16.48,0:07:18.55,Default,,0000,0000,0000,,Well, which pair would it be?
Dialogue: 0,0:07:18.55,0:07:24.04,Default,,0000,0000,0000,,He suggested that the number two\Nshould be the set of its predecessors.
Dialogue: 0,0:07:24.04,0:07:28.83,Default,,0000,0000,0000,,Two has two predecessors, zero and one.
Dialogue: 0,0:07:28.83,0:07:34.83,Default,,0000,0000,0000,,We take two to be the set\Nwhose members are zero and one.
Dialogue: 0,0:07:34.83,0:07:37.98,Default,,0000,0000,0000,,But we still have numbers;\Nwe have zero and one.
Dialogue: 0,0:07:37.98,0:07:42.75,Default,,0000,0000,0000,,Well, zero is the set of its predecessors.
Dialogue: 0,0:07:42.75,0:07:45.87,Default,,0000,0000,0000,,Zero has no predecessors, \Nso it's what's called the 'null set,'
Dialogue: 0,0:07:45.87,0:07:48.10,Default,,0000,0000,0000,,the set without any members.
Dialogue: 0,0:07:48.10,0:07:52.78,Default,,0000,0000,0000,,And one has one predecessor, \Nwhich is zero.
Dialogue: 0,0:07:52.78,0:07:57.27,Default,,0000,0000,0000,,So one is the set\Nwhose sole member is zero.
Dialogue: 0,0:07:57.28,0:08:01.73,Default,,0000,0000,0000,,So there we have two defined, \None defined and zero defined.
Dialogue: 0,0:08:01.73,0:08:05.68,Default,,0000,0000,0000,,If we put these definitions \Ntogether, we get the set.
Dialogue: 0,0:08:05.68,0:08:10.11,Default,,0000,0000,0000,,The number two is the set\Nwhose two members are the null set,
Dialogue: 0,0:08:10.11,0:08:11.85,Default,,0000,0000,0000,,which is the number zero,
Dialogue: 0,0:08:11.85,0:08:16.42,Default,,0000,0000,0000,,and the set whose sole member\Nis the null set, which is the number one.
Dialogue: 0,0:08:17.36,0:08:21.97,Default,,0000,0000,0000,,So that's according to von Neumann\Nwhat the number two is;
Dialogue: 0,0:08:21.97,0:08:24.30,Default,,0000,0000,0000,,it's sets all the way down
Dialogue: 0,0:08:25.11,0:08:27.20,Default,,0000,0000,0000,,- sets, not turtles -
Dialogue: 0,0:08:27.20,0:08:29.98,Default,,0000,0000,0000,,And you actually hit rock bottom, too.
Dialogue: 0,0:08:31.18,0:08:33.89,Default,,0000,0000,0000,,And similarly for all other numbers,
Dialogue: 0,0:08:33.89,0:08:37.16,Default,,0000,0000,0000,,the number three would be \Nan even more complicated thing, and so on.
Dialogue: 0,0:08:38.21,0:08:42.95,Default,,0000,0000,0000,,Remember, the Frege-Russell view\Ngave birth to monsters.
Dialogue: 0,0:08:42.95,0:08:46.00,Default,,0000,0000,0000,,Here, we no longer have a monster;
Dialogue: 0,0:08:46.00,0:08:47.94,Default,,0000,0000,0000,,the monster has turned into an angel,
Dialogue: 0,0:08:47.94,0:08:50.98,Default,,0000,0000,0000,,because although the number two\Ncontains other numbers,
Dialogue: 0,0:08:50.98,0:08:53.12,Default,,0000,0000,0000,,it doesn't contain itself.
Dialogue: 0,0:08:53.100,0:08:57.90,Default,,0000,0000,0000,,The monster is always eating \Na smaller monster, so to speak.
Dialogue: 0,0:08:58.91,0:09:00.78,Default,,0000,0000,0000,,It doesn't get in the way of itself.
Dialogue: 0,0:09:00.78,0:09:05.74,Default,,0000,0000,0000,,This view is generally accepted \Nby philosophers and mathematicians today,
Dialogue: 0,0:09:05.74,0:09:08.37,Default,,0000,0000,0000,,but it also has its difficulties.
Dialogue: 0,0:09:08.81,0:09:11.04,Default,,0000,0000,0000,,One difficulty that especially bothers me
Dialogue: 0,0:09:11.04,0:09:14.34,Default,,0000,0000,0000,,is there's nothing special\Nabout the number two.
Dialogue: 0,0:09:14.34,0:09:19.34,Default,,0000,0000,0000,,We want the number two \Nto be what is common to all pairs,
Dialogue: 0,0:09:20.02,0:09:25.25,Default,,0000,0000,0000,,but von Neumann's number two \Nis just one pair among many,
Dialogue: 0,0:09:25.25,0:09:26.86,Default,,0000,0000,0000,,and there's no special way
Dialogue: 0,0:09:26.86,0:09:30.69,Default,,0000,0000,0000,,in which that pair is\Nwhat's common to all pairs.
Dialogue: 0,0:09:31.80,0:09:34.39,Default,,0000,0000,0000,,So it doesn't make \Nthe number two special anyway;
Dialogue: 0,0:09:34.39,0:09:37.20,Default,,0000,0000,0000,,it's just one pair among many.
Dialogue: 0,0:09:37.20,0:09:42.51,Default,,0000,0000,0000,,We come now to the final view,\Nand the one I like most of all.
Dialogue: 0,0:09:44.75,0:09:49.74,Default,,0000,0000,0000,,It's a view that's generally \Ndismissed or ignored
Dialogue: 0,0:09:49.74,0:09:52.63,Default,,0000,0000,0000,,by philosophers \Nand mathematicians of today.
Dialogue: 0,0:09:52.63,0:09:58.44,Default,,0000,0000,0000,,It was developed by Georg Cantor\Nin the late 19th century.
Dialogue: 0,0:09:59.41,0:10:04.82,Default,,0000,0000,0000,,Cantor was a multi-talented individual, \N
Dialogue: 0,0:10:04.82,0:10:08.19,Default,,0000,0000,0000,,a brilliant violinist
Dialogue: 0,0:10:11.10,0:10:16.26,Default,,0000,0000,0000,,with wide-ranging interests\Nranging from religion to literature.
Dialogue: 0,0:10:17.44,0:10:20.91,Default,,0000,0000,0000,,But he's best known \Nfor his theory of infinite number.
Dialogue: 0,0:10:21.63,0:10:25.54,Default,,0000,0000,0000,,Cantor wanted to count \Nnot only finite collections
Dialogue: 0,0:10:25.54,0:10:29.43,Default,,0000,0000,0000,,- I know there are a lot of people here,\Nbut it's still a finite number -
Dialogue: 0,0:10:29.43,0:10:32.68,Default,,0000,0000,0000,,so not just the finite collections,\Nlike the number of people here,
Dialogue: 0,0:10:32.68,0:10:36.18,Default,,0000,0000,0000,,or the number of stars in the Milky Way,
Dialogue: 0,0:10:36.18,0:10:39.40,Default,,0000,0000,0000,,he also wanted\Nto count infinite collections,
Dialogue: 0,0:10:39.40,0:10:44.33,Default,,0000,0000,0000,,like the collection of all natural numbers\Nor the collection of all points in space.
Dialogue: 0,0:10:45.46,0:10:50.29,Default,,0000,0000,0000,,And to this end, he attempted \Nto develop a general theory of number.
Dialogue: 0,0:10:51.34,0:10:53.47,Default,,0000,0000,0000,,So what was his view?
Dialogue: 0,0:10:53.47,0:10:55.82,Default,,0000,0000,0000,,Again, let's consider the number two.
Dialogue: 0,0:10:55.82,0:10:59.70,Default,,0000,0000,0000,,Let's take two objects, Fido and Felix.
Dialogue: 0,0:11:00.26,0:11:01.61,Default,,0000,0000,0000,,Now Cantor said:
Dialogue: 0,0:11:01.61,0:11:08.20,Default,,0000,0000,0000,,"Look, let's deprive these two objects\Nof all of their individuating features
Dialogue: 0,0:11:08.20,0:11:11.70,Default,,0000,0000,0000,,beyond their being distinct\Nfrom one another."
Dialogue: 0,0:11:11.70,0:11:14.44,Default,,0000,0000,0000,,So we remove their fur,
Dialogue: 0,0:11:14.44,0:11:18.61,Default,,0000,0000,0000,,we remove their flesh and blood,
Dialogue: 0,0:11:18.61,0:11:22.06,Default,,0000,0000,0000,,until we're simply left \Nwith two bare objects
Dialogue: 0,0:11:22.06,0:11:25.48,Default,,0000,0000,0000,,- what he called units\Nwithout any differentiating features.
Dialogue: 0,0:11:25.48,0:11:28.21,Default,,0000,0000,0000,,I hope there are no animal lovers\Namongst you.
Dialogue: 0,0:11:28.21,0:11:32.74,Default,,0000,0000,0000,,But anyway, this is what happens to pets\Nwhen Cantor gets hold of them.
Dialogue: 0,0:11:33.97,0:11:36.20,Default,,0000,0000,0000,,So what are these units?
Dialogue: 0,0:11:36.20,0:11:40.03,Default,,0000,0000,0000,,Well, take the two dollars \Nin your bank account
Dialogue: 0,0:11:40.03,0:11:43.50,Default,,0000,0000,0000,,- I hope you still have two dollars left\Nafter paying the admission fee.
Dialogue: 0,0:11:43.50,0:11:47.36,Default,,0000,0000,0000,,These two dollars\Naren't any particular dollars,
Dialogue: 0,0:11:47.36,0:11:49.60,Default,,0000,0000,0000,,but when you go to the ATM machine,\N
Dialogue: 0,0:11:49.60,0:11:52.75,Default,,0000,0000,0000,,you can redeem them\Nfor two particular dollars.
Dialogue: 0,0:11:52.75,0:11:55.01,Default,,0000,0000,0000,,So they aren't those particular dollars,
Dialogue: 0,0:11:55.01,0:11:57.65,Default,,0000,0000,0000,,but they can be redeemed\Nfor any two particular dollars.
Dialogue: 0,0:11:57.65,0:11:59.70,Default,,0000,0000,0000,,This is what Cantor's units are like.
Dialogue: 0,0:11:59.70,0:12:03.81,Default,,0000,0000,0000,,But when you go to the Cantorian \NATM machine to redeem your units,
Dialogue: 0,0:12:03.81,0:12:06.59,Default,,0000,0000,0000,,you get back any two objects or whatever.
Dialogue: 0,0:12:06.59,0:12:08.87,Default,,0000,0000,0000,,It's the ultimate lucky dip.
Dialogue: 0,0:12:09.63,0:12:12.14,Default,,0000,0000,0000,,Cantor's idea was this: \N
Dialogue: 0,0:12:12.14,0:12:17.09,Default,,0000,0000,0000,,we take the number two\Nto be the set of these two units.
Dialogue: 0,0:12:17.09,0:12:19.23,Default,,0000,0000,0000,,So we take these two units,
Dialogue: 0,0:12:19.23,0:12:22.36,Default,,0000,0000,0000,,which could be derived\Nfrom any two objects,
Dialogue: 0,0:12:22.36,0:12:26.89,Default,,0000,0000,0000,,and the number two\Nis the set of those two units.
Dialogue: 0,0:12:26.89,0:12:29.02,Default,,0000,0000,0000,,And similarly, for all other numbers;
Dialogue: 0,0:12:29.02,0:12:31.41,Default,,0000,0000,0000,,the number three \Nwould be the set of three units,\N
Dialogue: 0,0:12:31.41,0:12:33.81,Default,,0000,0000,0000,,and so on and so forth.
Dialogue: 0,0:12:33.81,0:12:37.00,Default,,0000,0000,0000,,So we have three views on the table.
Dialogue: 0,0:12:37.00,0:12:38.81,Default,,0000,0000,0000,,The Frege-Russell view
Dialogue: 0,0:12:38.81,0:12:42.19,Default,,0000,0000,0000,,according to which the number two\Nis the set of all pairs;
Dialogue: 0,0:12:42.19,0:12:44.04,Default,,0000,0000,0000,,the view of von Neumann,
Dialogue: 0,0:12:44.04,0:12:47.98,Default,,0000,0000,0000,,according to which the number two\Nis the set whose members are zero and one;
Dialogue: 0,0:12:47.98,0:12:51.78,Default,,0000,0000,0000,,and the Cantorian view,
Dialogue: 0,0:12:51.78,0:12:55.59,Default,,0000,0000,0000,,according to which two\Nis the set of two units.
Dialogue: 0,0:12:55.59,0:13:01.52,Default,,0000,0000,0000,,The Frege-Russell view breeds \Nmonsters, so we can't have it.
Dialogue: 0,0:13:02.98,0:13:05.93,Default,,0000,0000,0000,,The von Neumann view \Ndoesn't properly account
Dialogue: 0,0:13:05.93,0:13:09.89,Default,,0000,0000,0000,,for why the number two \Nis common to all pairs.
Dialogue: 0,0:13:11.00,0:13:15.18,Default,,0000,0000,0000,,The Cantorian view suffers\Nfrom neither of those difficulties.
Dialogue: 0,0:13:15.18,0:13:18.100,Default,,0000,0000,0000,,It doesn't breed monsters because\Nthe number two only contains units;
Dialogue: 0,0:13:18.100,0:13:21.93,Default,,0000,0000,0000,,it doesn't itself contain the number two.
Dialogue: 0,0:13:21.93,0:13:25.10,Default,,0000,0000,0000,,And it's, in an obvious sense, \Ncommon to all pairs,
Dialogue: 0,0:13:25.10,0:13:28.14,Default,,0000,0000,0000,,because it's derived \Nby this process of abstraction,
Dialogue: 0,0:13:28.14,0:13:30.90,Default,,0000,0000,0000,,or stripping away, from each pair.
Dialogue: 0,0:13:33.00,0:13:36.87,Default,,0000,0000,0000,,So thanks to Cantor,\Nwe now know what numbers are.
Dialogue: 0,0:13:37.74,0:13:38.80,Default,,0000,0000,0000,,Thank you.
Dialogue: 0,0:13:38.80,0:13:40.09,Default,,0000,0000,0000,,(Applause)