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00:01There is a hole at the bottom of math.
00:04A hole that means we will never know everything with certainty.
00:09There will always be true statements that cannot be proven.
00:13Now, no one knows what those statements are exactly,
00:17but they could be something like the twin prime conjecture.
00:20Twin primes are prime numbers that are separated by just one number,
00:24like 11 and 13 or 17 and 19.
00:27And as you go up the number line, primes occur less frequently
00:31and twin primes are rarer still.
00:34But the twin prime conjecture is that there are infinitely many twin primes.
00:39You never run out.
00:41As of right now, no one has proven this conjecture true or false.
00:46But the crazy thing is this.
00:49We may never know.
00:51Because what has been proven is that in any system of mathematics
00:55where you can do basic arithmetic,
00:57there will always be true statements that are impossible to prove.
01:03That is life.
01:07Specifically, this is the game of life created in 1970 by mathematician John Conway.
01:13Sadly, he passed away in 2020 from COVID-19.
01:17Conway's game of life is played on an infinite grid of square cells,
01:22each of which is either live or dead.
01:25And there are only two rules.
01:27One, any dead cell with exactly three neighbors comes to life.
01:32And two, any living cell with less than two or more than three neighbors dies.
01:38Once you've set up the initial arrangement of cells,
01:42the two rules are applied to create the next generation.
01:45And then the one after that, and the one after that, and so on.
01:49It's totally automatic.
01:50Conway called it a zero player game.
01:53But even though the rules are simple,
01:55the game itself can generate a wide variety of behavior.
01:59Some patterns are stable.
02:01Once they arise, they never change.
02:03Others oscillate back and forth in a loop.
02:06A few can travel across the grid forever like this glider here.
02:11Many patterns just fizzle out.
02:17But a few keep growing forever.
02:21They keep generating new cells.
02:24Now you would think that given the simple rules of the game,
02:27you could just look at any pattern and determine what will happen to it.
02:31Will it eventually reach a steady state?
02:34Or will it keep growing without limit?
02:37But it turns out this question is impossible to answer.
02:42The ultimate fate of a pattern in Conway's game of life is undecidable.
02:47Meaning there is no possible algorithm that is guaranteed to answer the question in a finite amount of time.
02:53You could always just try running the pattern and see what happens.
02:57I mean the rules of the game are a kind of algorithm after all.
03:00But that's not guaranteed to give you an answer either.
03:03Because even if you run it for a million generations,
03:06you won't be able to say whether it will last forever or just two million generations.
03:11Or a billion.
03:12Or a Googleplex.
03:13Is there something special about the game of life that makes it undecidable?
03:18Nope.
03:19There are actually a huge number of systems that are undecidable.
03:24Like Wang tiles, quantum physics, airline ticketing systems, and even Magic the Gathering.
03:31To understand how undecidability shows up in all of these places,
03:36we have to go back 150 years to a full-blown revolt in mathematics.
03:42In 1874, Georg Cantor, a German mathematician, published a paper that launched a new branch of mathematics called set theory.
03:53A set is just a well-defined collection of things.
03:57So the two shoes on your feet are a set, as are all the planetariums in the world.
04:02There's a set with nothing in it, the empty set, and a set with everything in it.
04:07Now Cantor was thinking about sets of numbers, like natural numbers, positive integers like one, two, three, four, and so on.
04:15And real numbers, which include fractions like a third, five halves, and also irrational numbers like pi, e, and the square root of two.
04:25Basically, any number that can be represented as an infinite decimal.
04:30He wondered, are there more natural numbers, or more real numbers between zero and one?
04:38The answer might seem obvious.
04:40There are an infinite number of each, so both sets should be the same size.
04:44But to check this logic, Cantor imagined writing down an infinite list,
04:49matching up each natural number on one side with a real number between zero and one on the other.
04:54Now since each real number is an infinite decimal, there is no first one, so we can just write them down in any random order.
05:02The key is to make sure we get them all, with no duplicates, and line them up one to one with an integer.
05:10If we can do that with none left over, well then we know that the set of natural numbers and the set of real numbers between zero and one are the same size.
05:18So, assume we've done that. We have a complete infinite list with each integer acting like an index number,
05:26a unique identifier for each real number on the list.
05:29Now, Cantor says, start writing down a new real number.
05:34And the way we're going to do it is by taking the first digit of the first number and adding one.
05:40Then take the second digit of the second number and again add one.
05:44Take the third digit of the third number, add one, and keep doing this all the way down the list.
05:49If the digit is a nine, just roll it back to an eight.
05:52And by the end of this process, you'll have a real number between zero and one.
05:58But here's the thing. This number won't appear anywhere on our list.
06:03It's different from the first number in the first decimal place, different from the second number in the second decimal place,
06:08and so on down the line.
06:10It has to be different from every number on the list by at least one digit, the number on the diagonal.
06:19That's why this is called Cantor's diagonalization proof.
06:22It shows there must be more real numbers between zero and one than there are natural numbers extending out to infinity.
06:31So not all infinities are the same size.
06:35Cantor called these countable and uncountable infinities respectively.
06:39And in fact, there are many more uncountable infinities which are even larger.
06:44Now Cantor's work was just the latest blow to mathematics.
06:47For 2000 years, Euclid's elements were considered the bedrock of the discipline.
06:52But at the turn of the 19th century, Loboshevsky and Gauss discovered non-Euclidean geometries.
06:58And this prompted mathematicians to examine more closely the foundations of their field.
07:03And they did not like what they saw.
07:06The idea of a limit at the heart of calculus turned out to be poorly defined.
07:11And now Cantor was showing that infinity itself was much more complex than anyone had imagined.
07:16In all this upheaval, mathematics fractured and a huge debate broke out among mathematicians at the end of the 1800s.
07:24On the one side were the intuitionists who thought that Cantor's work was nonsense.
07:29They were convinced that math was a pure creation of the human mind and that infinities like Cantor's weren't real.
07:37Henri Poincaré said that later generations will regard set theory as a disease from which one has recovered.
07:44Leopold Kroniker called Cantor a scientific charlatan and a corrupter of the youth.
07:50And he worked to keep Cantor from getting a job he wanted.
07:53On the other side were the formalists.
07:55They thought that math could be put on absolutely secure logical foundations through Cantor's set theory.
08:01The informal leader of the formalists was the German mathematician David Hilbert.
08:06Hilbert was a living legend, a hugely influential mathematician who had worked in nearly every area of mathematics.
08:13He almost beat Einstein to the punch on general relativity.
08:17He developed entirely new mathematical concepts that were crucial for quantum mechanics.
08:21And he knew that Cantor's work was brilliant.
08:24Hilbert was convinced that a more formal and rigorous system of mathematical proof based on set theory could solve all the issues that had cropped up in math over the last century.
08:34And most other mathematicians agreed with him.
08:36No one shall expel us from the paradise that Cantor has created, Hilbert declared.
08:42But in 1901, Bertrand Russell pointed out a serious problem in Cantor's set theory.
08:47Russell knew that if sets can contain anything, they can contain other sets or even themselves.
08:53For example, the set of all sets must contain itself, as does the set of sets with more than five elements in them.
09:00You could even talk about the set of all sets that contain themselves.
09:04But this leads straight to a problem.
09:07What about R, the set of all sets that don't contain themselves?
09:12If R doesn't contain itself, well, then it must contain itself.
09:17But if R does contain itself, then by definition, it must not contain itself.
09:21So R contains itself if and only if it doesn't.
09:26Russell had found another paradox of self-reference.
09:30And he later explained his paradox using a hairy analogy.
09:34Let's say there's a village populated entirely by grown men with a strange law against beards.
09:40Specifically, the law states that the village barber must shave all and only those men of the village who do not shave themselves.
09:47But the barber himself lives in the village too, of course, and he's a man.
09:52So who shaves him?
09:54If he doesn't shave himself, then the barber has to shave him.
09:58But the barber can't shave himself because the barber doesn't shave anyone who shaves themselves.
10:03So the barber must shave himself if and only if he doesn't shave himself.
10:08It's a contradiction.
10:11The intuitionists rejoiced at Russell's paradox, thinking it had proven set theory hopelessly flawed.
10:17But Zermelo and other mathematicians from Hilbert School solved the problem by restricting the concept of a set.
10:23So the collection of all sets, for example, is not a set anymore.
10:27And neither is the collection of all sets which don't contain themselves.
10:31This eliminated the paradoxes that come with self-reference.
10:34Hilbert and the formalists lived to fight another day, but self-reference refused to die quite that easily.
10:43Fast forward to the 1960s, and mathematician Hao Wang was looking at square tiles with different colors on each side like these.
10:51The rules were that touching edges must be the same color, and you can't rotate or reflect tiles.
10:56Only slide them around.
10:58The question was, if you're given an arbitrary set of these tiles, can you tell if they will tile the plane?
11:04That is, will they connect up with no gaps all the way out to infinity?
11:09It turns out, you can't tell for an arbitrary set of tiles whether they will tile the plane or not.
11:16The problem is undecidable.
11:19Just like the fate of a pattern in Conway's Game of Life.
11:22In fact, it's exactly the same problem.
11:25And that problem, ultimately, comes from self-reference, as Hilbert and the formalists were about to discover.
11:32Hilbert wanted to secure the foundations of mathematics by developing a new system for mathematical proofs.
11:39Systems of proof were an old idea going back to the ancient Greeks.
11:43A system of proof starts with axioms, basic statements that are assumed to be true, like a straight line can be drawn between any two points.
11:51Proofs are then constructed from those axioms using rules of inference, methods for using existing statements to derive new statements.
11:59And these are chosen to preserve truth.
12:02If the existing statements are true, then so are the new ones.
12:05Hilbert wanted a formal system of proof.
12:08A symbolic, logical language with a rigid set of manipulation rules for those symbols.
12:13Logical and mathematical statements could then be translated into this system.
12:17If you drop a book, then it will fall would be A then B.
12:22And no human is immortal would be expressed like this.
12:26Hilbert and the formalists wanted to express the axioms of mathematics as symbolic statements in a formal system,
12:34and set up the rules of inference as the system's rules for symbol manipulation.
12:39So Russell, along with Alfred North Whitehead, developed a formal system like this in their three-volume,
12:45Principia Mathematica, published in 1913.
12:48Principia Mathematica is vast, a total of nearly 2,000 pages of dense mathematical notation.
12:55It takes 762 pages just to get to a complete proof that 1 plus 1 equals 2.
13:02At which point Russell and Whitehead dryly note,
13:05the above proposition is occasionally useful.
13:09The authors had originally planned a fourth volume,
13:12but unsurprisingly, they were too worn out to complete it.
13:16So yes, the notation is dense and exhausting, but it is also exact,
13:21unlike ordinary languages.
13:23It leaves no room for errors or fuzzy logic to creep in.
13:27And most importantly, it allows you to prove properties of the formal system itself.
13:36There were three big questions that Hilbert wanted answered about mathematics.
13:40Number one, is math complete?
13:43Meaning, is there a way to prove every true statement?
13:48Does every true statement have a proof?
13:50Number two, is mathematics consistent?
13:53Meaning, is it free of contradictions?
13:56I mean, if you can simultaneously prove A and not A,
14:00then that's a real problem because you can prove anything at all.
14:04And number three, is math decidable?
14:08Meaning, is there an algorithm that can always determine whether a statement follows from the axioms?
14:15Now, Hilbert was convinced that the answers to all three of these questions was yes.
14:22At a major conference in 1930, Hilbert gave a fiery speech about these questions.
14:28He ended it with a line that summed up his formalist dream.
14:31In opposition to the foolish ignorabimus, which means we will not know,
14:36our slogan shall be, we must know, we will know.
14:41These words are literally on his grave.
14:44But by the time Hilbert gave this speech, his dream was already crumbling.
14:50Just the day before, at a small meeting at the same conference,
14:53a 24-year-old logician named Kurt Gödel explained that he had found the answer
14:58to the first of Hilbert's three big questions about completeness.
15:01And the answer was no.
15:04A complete, formal system of mathematics was impossible.
15:08The only person who paid much attention was John von Neumann,
15:11one of Hilbert's former students, who pulled Gödel aside to ask a few questions.
15:15But the next year, Gödel published a proof of his incompleteness theorem,
15:19and this time, everyone, including Hilbert, took notice.
15:29This is how Gödel's proof works.
15:31Gödel wanted to use logic and mathematics to answer questions
15:37about the very system of logic and mathematics.
15:42And so he took all these basic symbols of a mathematical system,
15:47and then he gave each one a number.
15:51This is known as the symbol's Gödel number.
15:54So, the symbol for not gets the number one,
15:58or has the Gödel number two,
16:01if then is Gödel number three.
16:04Now, if you're expressing all of these symbols with numbers,
16:07then what do you do about the numbers themselves?
16:10Well, zero gets its own Gödel number, six,
16:13and if you want to write a one,
16:15you just put this successor symbol next to it.
16:18The immediate successor of zero is one.
16:21And if you want to write two,
16:23then you have to write ss0,
16:26and that represents two, and so on.
16:28So you could represent any positive integer this way.
16:31Granted, it is cumbersome, but it works,
16:34and that is the point of this system.
16:36So, now that we have Gödel numbers for all of the basic symbols
16:40and all of the numbers we might want to use,
16:43we can start to write equations.
16:45Like, we could write zero equals zero.
16:49So, these symbols have the Gödel numbers six, five, six.
16:55And we can actually create a new card
16:58that represents this equation, zero equals zero.
17:02And the way we do it is we take the prime numbers,
17:06starting at two,
17:07and we raise each one to the power of the Gödel number
17:11of the symbol in our equation.
17:13So, we have two to the power of six
17:15times three to the power of five
17:17times five to the power of six equals 243 million.
17:21So, 243 million is the Gödel number
17:25of the whole equation zero equals zero.
17:30We can write down Gödel numbers
17:32for any set of symbols that you can imagine.
17:36So, really, this is an infinite deck of cards
17:39where any set of symbols that you want to write down
17:42in a sequence you can represent with a unique Gödel number.
17:47And the beauty of this Gödel number is
17:49you can do a prime factorization of it
17:51and you can work out exactly
17:53what symbols made up this card.
17:57So, in this whole pack of cards
17:59there are going to be true statements
18:01and there are going to be false statements.
18:03So, how do you prove that something is true?
18:06Well, you need to go to the axioms.
18:09The axioms also have their own Gödel numbers
18:13which are formed in the same way.
18:15So, here's an axiom which says
18:17not the successor of any number x equals zero.
18:23Which makes sense because in this system
18:25there are no negative numbers
18:27and so the successor of any number cannot be zero.
18:30So, we can put down this axiom
18:32and then we can substitute in zero for x
18:36saying that one does not equal zero.
18:39and we've created a proof.
18:41This is the simplest proof I can think of
18:45that shows that one does not equal zero.
18:48Now, this card for the proof of one does not equal zero
18:52gets its own Gödel number.
18:54And the way we calculate this Gödel number
18:56is just as before we take the prime numbers
18:59and we raise two to the power of the axiom times three
19:04to the power of one does not equal zero.
19:07And we get a tremendously large number.
19:10It would have 73 million digits if you calculated it all out.
19:14So, I've just left it here in exponential notation.
19:17As you can see, these numbers get very large
19:21and so you might want to start just calling them by letters.
19:25So, we could say this one is Gödel number A,
19:28this is Gödel number B, Gödel number C, and so on.
19:32Gödel goes to all this trouble to find this card
19:38which says there is no proof for the statement with Gödel number G.
19:44The trick is that the Gödel number of this card is G.
19:51So, what this statement is really saying is this card is unprovable.
19:56There is no proof anywhere in our infinite deck for this card.
20:01Now, think about it.
20:02If it's false and there is a proof,
20:05then what you have just proven is there is no proof.
20:09So, you're stuck with a contradiction.
20:12That would mean the mathematical system is inconsistent.
20:18The other alternative is that this card is true.
20:22There is no proof for the statement with Gödel number G.
20:25But that means this mathematical system has true statements in it that have no proof.
20:32So, your mathematical system is incomplete.
20:35And that is Gödel's incompleteness theorem.
20:39This is how he shows that any basic mathematical system that can do fundamental arithmetic
20:45will always have statements within it which are true but which have no proof.
20:52There is a line in the TV show The Office that echoes the self-referential paradox of Gödel's proof.
20:58Jim is my enemy, but it turns out that Jim is also his own worst enemy.
21:02And the enemy of my enemy is my friend.
21:05So, Jim is actually my friend.
21:08But, because he is his own worst enemy, the enemy of my friend is my enemy.
21:14So, actually, Jim is my enemy.
21:16Goodell's incompleteness theorem means that truth and provability are not at all the same thing.
21:29Hilbert was wrong.
21:31There will always be true statements about mathematics that just cannot be proven.
21:35Now, Hilbert could console himself with the hope that at least we could still prove maths consistent,
21:40that is, free of contradictions, but then Gödel published his second incompleteness theorem
21:46in which he showed that any consistent formal system of math cannot prove its own consistency.
21:52So, taken together, Gödel's two incompleteness theorems say that the best you can hope for
21:58is a consistent yet incomplete system of math, but a system like that cannot prove its own consistency.
22:05So, some contradiction could always crop up in the future,
22:08revealing that the system you'd been working with had been inconsistent the whole time.
22:13And that leaves only the third and final big question from Hilbert.
22:17Is mathematics decidable?
22:19That is, is there an algorithm that can always determine whether a statement follows from the axioms?
22:24And in 1936, Alan Turing found a way to settle this question.
22:29But, to do it, he had to invent the modern computer.
22:33In his time, computers weren't machines.
22:37They were people, often women who carried out long and tedious calculations.
22:42Turing imagined an entirely mechanical computer.
22:45He wanted it to be powerful enough to perform any computation imaginable,
22:49but simple enough that you could reason through its operation.
22:53So, he came up with a machine that takes as input an infinitely long tape of square cells,
22:59each containing a zero or a one.
23:02The machine has a read-write head that can read one digit at a time.
23:07And then it can perform one of only a few tasks.
23:10Overwrite a new value.
23:12Move left.
23:13Move right.
23:14Or simply, halt.
23:16Halting means the program has run to completion.
23:20Now, the program consists of a set of internal instructions.
23:24You can think of it like a flow chart that tells the machine what to do based on the digit it reads and its internal state.
23:32You could imagine exporting these instructions to any other Turing machine,
23:36which would then perform in exactly the same way as the first.
23:39Although this sounds simple, a Turing machine's arbitrarily large memory and program mean it can execute any computable algorithm if given enough time.
23:50From addition and subtraction to the entire YouTube algorithm, it can do anything a modern computer can.
23:57That's what made the Turing machine so useful in answering Hilbert's question on decidability.
24:03When a Turing machine halts, the program has finished running and the tape is the output.
24:08But sometimes a Turing machine never halts.
24:12Maybe it gets stuck in an infinite loop.
24:14Is it possible to tell beforehand if a program will halt or not on a particular input?
24:20Turing realized this halting problem was very similar to the decidability problem.
24:25If you could find a way to figure out if a Turing machine would halt,
24:28then it would also be possible to decide if a statement followed from the axioms.
24:33For example, you could solve the twin prime conjecture by writing a Turing machine program that starts with the axioms
24:39and constructs all theorems that can be produced in one step using the rules of inference.
24:44Then it constructs all theorems that can be produced in one step from those, and so on.
24:49Each time it generates a new theorem, it checks if it's the twin prime conjecture.
24:53And if it is, it halts. If not, it never halts.
24:57So if you could solve the halting problem, you could solve the twin prime conjecture and all sorts of other unsolved questions.
25:04So Turing said, let's assume we can make a machine H that can determine whether any Turing machine will halt or not on a particular input.
25:14You insert the program and its input, and H simulates what will happen, printing out either halts or never halts.
25:22For now, we don't worry about how H works. We just know that it always works. It always gives you the right answer.
25:29Now, we can modify the H machine by adding additional components.
25:34One, if it receives the output halts, immediately goes into an infinite loop.
25:40Another, if it receives never halts, then it immediately halts.
25:45We can call this entire new machine H+, and we can export the program for that entire machine.
25:54Now, what happens if we give this machine its own code both as program and input?
26:02Well now, H is simulating what H plus would do given its own input.
26:08Essentially, H has to determine the behavior of a machine that it itself is a part of in this exact circumstance.
26:17If H concludes that H plus never halts, well, this makes H plus immediately halt.
26:29If H thinks H plus will halt, well then that necessarily forces H plus to loop.
26:39Whatever output the halting machine H gives, it turns out to be wrong.
26:45There's a contradiction.
26:47The only explanation can be that a machine like H can't exist.
26:52There's no way to tell, in general, if a Turing machine will halt or not on a given input.
26:58And this means mathematics is undecidable.
27:02There is no algorithm that can always determine whether a statement is derivable from the axioms.
27:08So something like the twin prime conjecture might be unsolvable.
27:12We might never know whether there are infinite twin primes or not.
27:17The problem of undecidability even crops up in physical systems.
27:22In quantum mechanics, one of the most important properties of a many-body system is the difference in energy between its ground state and its first excited state.
27:31This is known as the spectral gap.
27:34Some systems have a significant spectral gap, whereas others are gapless.
27:39There are a continuum of energy levels all the way down to the ground state.
27:43And this is important because at low temperatures, gapless quantum systems can undergo phase transitions while gapped systems cannot.
27:51They don't have the energy needed to overcome the spectral gap.
27:55But figuring out if a system is gapped or gapless has long been known to be a very difficult problem.
28:03Only recently, in 2015, did mathematicians prove that in general the spectral gap question is undecidable.
28:12To quote the authors,
28:14Even a perfect complete description of the microscopic interactions between a material's particles is not always enough to deduce its macroscopic properties.
28:24Now remember that Turing designed his machines to be as powerful computers as it is possible to make.
28:30To this day, the best computational systems are those that can do everything a Turing machine can.
28:36This is called Turing completeness.
28:38And it turns out, there are many such Turing complete systems.
28:42Although they are powerful, every Turing complete system comes with a catch.
28:47Its own analog of the halting problem.
28:50Some undecidable property of the system.
28:52Wang tiles are Turing complete.
28:55Their halting problem is whether or not they'll tile the plane.
28:58Complex quantum systems are Turing complete.
29:01And their halting problem is the spectral gap question.
29:05And the game of life is Turing complete.
29:07Its halting problem is literally whether or not it halts.
29:11There are many more examples of this.
29:14Like airline ticketing systems, magic the gathering, powerpoint slides, and excel spreadsheets.
29:19Nearly every programming language in existence is designed to be Turing complete.
29:24But in theory, we only need one programming language.
29:27Because you can program anything at all using any Turing complete system.
29:32So here's a Turing machine inside the game of life.
29:41And since the game of life is itself Turing complete, it should be capable of simulating itself.
29:47And indeed, it is.
29:49The real legacy of David Hilbert's dream is all of our modern computational devices.
30:15Kurt Gödel suffered bouts of mental instability later in life.
30:20Convinced that people were trying to poison him, he refused to eat.
30:25Eventually, starving himself to death.
30:28Hilbert died in 1943.
30:31His epitaph was his slogan from 1930.
30:34We must know, we will know.
30:36The truth is, we don't know.
30:39Sometimes, we can't know.
30:42But in trying to find out, we can discover new things.
30:46Things that change the world.
30:48Alan Turing put his ideas about computing to practical use in World War II.
30:53Leading the team at Bletchley Park that built real calculating machines to crack Nazi codes for the Allies.
30:58By one estimate, the intelligence Turing and his colleagues gathered from decrypted messages shortened the war by two to four years.
31:07After the war, Turing and John von Neumann designed the first true programmable electronic computer, ENIAC, based on Turing's designs.
31:16But Turing didn't live to see his ideas develop much further.
31:20In 1952, the British government convicted him of gross indecency upon learning he was gay.
31:26He was stripped of his security clearance and forced to take hormones.
31:29In 1954, he committed suicide.
31:36Turing changed the world.
31:38He is widely considered to be the most important founding figure in computer science.
31:42All modern computers are descended from his designs.
31:46But Turing's ideas about computability came from his concept of the Turing machine.
31:51Which came from thinking about Hilbert's question, is math decidable?
31:56So, Turing's code-breaking machines, and indeed all modern computers, stem from the weird paradoxes that arise from self-reference.
32:07There is a hole at the bottom of math.
32:10A hole that means we will never know everything with certainty.
32:14There will always be true statements that cannot be proven.
32:18And you might think this realization would have driven mathematicians mad,
32:22and led to the disintegration of the entire mathematical enterprise.
32:25But instead, thinking about this problem transformed the concept of infinity,
32:31changed the course of a world war,
32:33and led directly to the invention of the device you're watching this on right now.
32:38Hey, this video was sponsored by Brilliant.
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33:49Just go to brilliant.org slash Veritasium,
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33:57and I want to thank you for watching.
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