Shadowing Practice: The Banach–Tarski Paradox - Learn English Speaking with Video
Creando lección...
1
Hey Vsauce, Michael here.
2
There's a famous way to seemingly create chocolate out of nothing.
3
Maybe you've seen it before.
4
This chocolate bar is four squares by eight squares.
5
But if you cut it like this, and then like this, and finally like this,
6
you can rearrange the pieces like so and wind up with the same four by eight bar,
7
but with a leftover piece apparently created out of thin air.
8
There's a popular animation of this illusion as well.
9
I call it an illusion because it's just that, fake.
10
In reality, the final bar is a bit smaller.
11
It contains this much less chocolate.
12
Each square along the cut is shorter than it was in the original, but the cut makes it difficult to notice right away.
13
The animation is extra misleading because it tries to cover up its deception.
14
The lost height of each Each square is surreptitiously added in while the piece moves to make it hard to notice.
15
I mean, come on.
16
Obviously, you cannot cut up a chocolate bar and rearrange the pieces into more than you started with.
17
- Okay.
18
Or can you?
19
One of the strangest theorems in modern mathematics is the Banach -Tarski paradox.
20
It proves that there is in fact a way to take an object and separate it into five different pieces. of the original item.
21
Same density, same size, same everything.
22
Seriously, to dive into the mind blow that it is and the way it fundamentally questions math and ourselves,
23
we have to start by asking a few questions.
24
First, what is infinity?
25
A number?
26
I mean, it's nowhere on the number line, but we often say things like, "There's an infinite number of blah blah blah.
27
And as far as we know, infinity could be real.
28
The universe may be infinite in size and flat, extending out forever and ever without end,
29
beyond even the part we can't observe or ever hope to observe.
30
That's exactly what infinity is.
31
Not a number per se, but rather a size.
32
the size of something that doesn't end.
33
Infinity is not the biggest number.
34
Instead, it is how many numbers there are.
35
But there are different sizes of infinity.
36
The smallest type of infinity is countable infinity, the number of hours in forever.
37
It's also the number of whole numbers that there are, natural numbers, the numbers we use when counting things like 1,
38
2, 3, 4, 5, 6, and so on.
39
Sets like these are unending, but they are countable.
40
Countable means that you can count them from one element to any other in a finite amount of time.
41
Even if that finite amount of time is longer than you will live, or the universe will exist for, it's still finite.
42
Uncountable infinity, on the other hand, is literally bigger.
43
Too big to even count.
44
The number of real numbers that there are, not just whole numbers, but all numbers, is uncountably infinite.
45
You literally cannot count even from zero to one in a finite amount of time by naming every real number in between.
46
I mean, where do you even start?
47
Zero?
48
Okay, but What comes next?
49
0 .000000.
50
Eventually, we would imagine a 1 going somewhere at the end, but there is no end.
51
We could always add another 0.
52
Uncountability makes this set so much larger than the set of all whole numbers
53
that even between 0 and 1 there are more numbers than there are whole numbers on the entire endless number line.
54
Derokantor's famous diagonal argument helps illustrate this.
55
Imagine listing every number between 0 and 1.
56
Since they are uncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats.
57
Bye.
58
Each number we generate can be paired with a whole number.
59
If there's a one -to -one correspondence between the two, that is if we can match one whole number to each real number on our list,
60
that would mean that countable and uncountable sets are the same size.
61
But we can't do that.
62
Even though this list goes on forever, Forever isn't enough.
63
Watch this.
64
If we go diagonally down our endless list of real numbers and take the first decimal of the first number, and the second of the second number,
65
the third of the third, and so on, and add one to each, subtracting one if it happens to be a nine,
66
we can generate a new real number that is obviously between 0 and 1,
67
but since we've defined it to be different from every number on our endless list in at least one place,
68
It's clearly not contained in the list.
69
In other words, we've used up every single whole number, the entire infinity of them,
70
and yet we can still come up with more real numbers.
71
Here's something else that is true but counterintuitive.
72
There are the same number of even numbers as there are even and odd numbers.
73
At first, that sounds ridiculous.
74
Clearly there are only half as many even numbers as all whole numbers.
75
But that intuition is wrong.
76
The set of all whole numbers is denser, but every even number can be matched with a whole number.
77
You will never run out of members of either set, so this one -to -one correspondence shows that both sets are the same size.
78
In other words, infinity divided by two is still infinity.
79
Infinity plus one is also infinity. of the Grand Hotel.
80
Imagine a hotel with a countably infinite number of rooms.
81
But now, imagine that there is a person booked into every single room.
82
Seemingly, it's fully booked.
83
Right?
84
No. Infinite sets go against common sense.
85
You see, if a new guest shows up and wants a room, all the hotel has to do is move the guest in room number one to room number two.
86
And the guest in room two to room three and three to four and four to five and so on.
87
Because the number of rooms is never ending, we cannot run out of rooms.
88
Infinity minus one is also infinity again.
89
If one guest leaves the hotel, we can shift every guest the other way.
90
Guest two goes to room one, three to two, four to three, and so on.
91
Because we have an infinite amount of guests, that is a never -ending supply of them, no room will be left empty As it turns out,
92
you can subtract any finite number from infinity and still be left with infinity.
93
It doesn't care.
94
It's unending.
95
Banoktarski hasn't left our sights yet.
96
All of this is related.
97
We are now ready to move on to Shapes.
98
Hilbert's Hotel can be applied to a circle.
99
Points around the circumference can be thought of as guests.
100
If we remove one point from the circle, That point is gone.
101
Right?
102
Infinity tells us it doesn't matter.
103
The circumference of a circle is irrational.
104
It's the radius times 2 pi.
105
So if we mark off points beginning from the hole, every radius length along the circumference going clockwise,
106
we will never land on the same point twice.
107
Ever.
108
has checked out, we can just shift the rest, move them counterclockwise, and every room will be filled.
109
Point 1 moves to fill in the hole.
110
Point 2 fills in the place where point 1 used to be.
111
3 fills in 2 and so on.
112
Since we have an unending supply of numbered points, No hole will be left unfilled.
113
The missing point is forgotten.
114
we apparently never needed it to be complete.
115
There's one last neato consequence of infinity we should discuss before tackling Bonnachtarski.
116
Ian Stewart famously proposed a brilliant dictionary, one that he called the Hyper Webster.
117
The Hyper Webster lists every single possible word of any length formed from the 26 letters in the English alphabet.
118
It begins with A followed by AA, then AAA, then AAAA, and after an infinite number of those,
119
AB then ABA, then ABAA, ABAAA, and so on until Z.
120
ZA, ZAA, etc, etc, until the final entry, an infinite sequence of Zs.
121
Such a dictionary would contain every single word.
122
Nay, every single thought, definition, description, truth, lie, name, story, what happened to Amelia Earhart would be in that dictionary.
123
as well as every single thing that didn't happened to Amelia Earhart.
124
everything that could be said using our alphabet.
125
Obviously, it would be huge.
126
But the company publishing it might realize that they could take a shortcut.
127
If they put all the words that begin with "A" in a volume titled "A", they wouldn't have to print the initial "A".
128
Readers would know to just add the A because it's the A volume.
129
By removing the initial A, the publisher is left with every A word Sans the first A.
130
which has surprisingly become every possible word.
131
Just one of the 26 volumes has been decomposed into the entire thing.
132
It is now.
133
that we're ready to investigate this video's titular paradox.
134
What if we turned an object, a 3D thing, into a Hyper Webster?
135
Could we decompose pieces of it into the whole thing?
136
Yes.
137
The first thing we need to do is give every single point on the surface of the sphere, one name and one name only.
138
A good way to do this is to name them after how they can be reached by a given starting point.
139
If we move this starting point across the surface of the sphere in steps that are just the right length,
140
no matter how many times or in what direction we rotate, So long as we never backtrack, it will never wind up in the same place twice.
141
We only need to rotate in four directions to achieve this paradox: up, down, left and right, around two perpendicular axes.
142
We are going to need every single possible sequence
143
that can be made of any finite length out of just these four rotations.
144
That means we will need left, right, up and down, as well as left left,
145
left up, left down, but of course not left right because, well, that's backtracking.
146
Going left and then right means you're the same as you were before you did anything.
147
So, no left rights, no right lefts, and no up downs, and no down ups.
148
Also, notice that I'm writing the rotations in order right to left.
149
So the final rotation is the leftmost letter.
150
That will be important later on.
151
Anyway, a list of all possible sequences of allowed rotations that are finite in length, is Well, huge.
152
countably infinite, in fact.
153
But if we apply each one of them to a starting point, in green here, and then name the point we land on after the sequence that brought us there,
154
we can name a countably infinite set of points on the surface.
155
Let's look at how, say, these four strings on our list would work.
156
right up left.
157
Okay, rotating the starting point this way takes us here.
158
Let's color code the point based on the final rotation in its string.
159
In this case, it's left and and for that we will use purple.
160
Next up, "Down, down." that sequence takes us here.
161
We name the point DD and color it blue since we ended with a down rotation.
162
RDR, that will be this point's name, takes us here, and for a final right rotation, let's use red.
163
Finally, for a sequence that ends with "up", let's color code the point.
164
Orange.
165
Now, if we imagine completing this process for every single sequence, we will have a countably infinite number of points named and color -coded.
166
That's great!
167
But not enough there are an uncountably infinite number of points on a spheres.
168
surface.
169
But no worries, we can just pick a point we missed.
170
any point and color it green, making it a new starting point, and then run every sequence from here.
171
After doing this, to an uncountably infinite number of starting points, we will have indeed named and colored every single point on the surface.
172
just once.
173
with the exception of Poles.
174
Every sequence has two poles of rotation, locations on the sphere that come back to exactly where they started.
175
for any sequence of right, or left rotations, the poles are the north and south poles.
176
The problem with poles like these is
177
that more than one sequence can lead us to them they can be named more than once
178
and be colored in more than one color.
179
For example, if you follow some other sequence to the North or South Pole, any subsequent rights or lefts will be equally valid names.
180
In order to deal with this, we're going to just count them out of the normal scheme and color them all yellow.
181
Every sequence has two, so there are a countably infinite amount of them.
182
Now, with every point on the sphere given just one name, in just one of six colors we are ready to take the entire sphere apart.
183
Every point on the surface corresponds to a unique line of points below it,
184
all the way to the center point and we will be dragging every points line along with it.
185
The lone center point we will set aside.
186
Okay, first we cut out and extract all the yellow poles.
187
the green starting points, the orange up points the blue down points, and the red and purple left and right points.
188
That's the entire sphere.
189
With just these pieces, you could build the whole thing.
190
But take a look at the left piece.
191
It is defined by being a piece composed of every point accessed via a sequence ending with a left rotation.
192
If we rotate this piece right, that's the same as adding an R to every point's name.
193
But left and then right is a backtrack.
194
They cancel each other out.
195
And look what happens when we reduce them away.
196
The set becomes the same as a set of all points with names that end with L,
197
but also U, D, and every point reached with no rotation.
198
That's the full set of starting points.
199
We have turned less than a quarter of the sphere into nearly three quarters.
200
Just by rotating it, we added nothing.
201
It's like the Hyper Webster.
202
If we add the right piece and the poles of rotation in the center point, well, we've got the entire sphere again, but with stuff left over.
203
To make a second copy, let's rotate the up piece down.
204
The down -ups cancel because, well, it's the same as going nowhere, and we're left with a set of all starting points,
205
the entire up piece, the right piece and the left piece But there's a problem here.
206
We don't need this extra set of starting points.
207
We still haven't used the original ones.
208
No worries.
209
Let's just start over we can just move everything from the up piece that turns into a starting point when rotated down.
210
That means every point whose final rotation is up.
211
Let's put them in the down piece.
212
Of course, after rotating, points named UU will just turn into points named U.
213
And that would give us a copy here and here.
214
So, as it turns out, we need to move all points with any name that is just a string of U's.
215
We will put them in the down piece and rotate the up piece down, which makes it congruent to the up, right, and left pieces.
216
Add in the down piece along with some up and the starting point piece and, well, we're almost done.
217
Rotation and center are missing from this copy, but no worries.
218
There's a countably infinite number of holes where the poles of rotation used to be.
219
Which means there is some pole around which we can rotate this sphere such that every pole hole orbits around
220
without hitting another.
221
Well, this is just a bunch of circles with one point missing.
222
We fill them each like we did earlier.
223
And we do the same for the center point.
224
Imagine a circle that contains it inside the sphere and just fill in from infinity.
225
And look what we've done.
226
We have taken one sphere, and turned it into two identical spheres without adding Anything.
227
1 plus 1 equals 1.
228
Thank you.
229
That took a while to go through, but the implications are huge and mathematicians, scientists, and philosophers are still debating them.
230
Could such a process happen in the real world I mean, it can happen mathematically, and math allows us to abstractly predict
231
and describe a lot of things in the real world with amazing accuracy.
232
But does the Banach -Tarski paradox take it too far?
233
Is it a place where math and physics Separate.
234
We still don't know.
235
History is full of examples of mathematical concepts developed in the abstract that we
236
did not think would ever apply to the real world for years,
237
decades, centuries, until Eventually, science caught up and realized they were totally applicable and useful,
238
The Bonnachtarski paradox could actually happen in our real world.
239
The only catch, of course, is that the five pieces you cut your object into aren't simple shapes,
240
they must be infinitely complex and detailed That's not possible to do in the real world, where measurements can only get so small,
241
and there's only a finite amount of time to do anything.
242
but math says it's theoretically valid and some scientists think it may be physically valid 2.
243
There have been a number of papers published suggesting a link between Banach Tarski,
244
and the way tiny, tiny subatomic particles can collide.
245
high energies and turn into more particles than we began with.
246
We are finite creatures.
247
Our lives are small, and can only scientifically consider a small part of reality,
248
What's common for us is just a sliver of what's available.
249
We can only see so much of the electromagnetic spectrum.
250
We can only delve so deep into extensions of space.
251
Common sense applies to that which we can access.
252
But common sense is just that.
253
"common." If total sense is what we want, we should be prepared to accept that we shouldn't call infinity weird or strange.
254
The results we've arrived at by accepting it are valid, true within the system we use to understand, measure, predict, and order the universe.
255
Perhaps the system still needs perfecting.
256
But at the end of the day, history continues to show us that the universe isn't strange.
257
We are.
258
And as always, thanks for watching there that really helped me wrap my mind kind of, around Banat Tarski.
259
First of all, Leonard Wapner's The Pea and The Sun.
260
This book is fantastic and it's full of a lot of the preliminaries needed to understand the proof that comes later.
261
He also talks a lot about the ramifications of what Bonnachtarski and their theorem might mean for mathematics.
262
Also, if you want to talk about math and whether it's discovered or invented, whether it really truly will map onto the universe, .
263
Yanofsky's...
264
The Outer Limits of Reason is great.
265
This is the favorite book of mine that I've read this entire year.
266
Another good one is E.
267
Brian Davies.
268
Why Beliefs Matter.
269
This is actually Korn's favorite book as you might be able to see there.
270
It's delicious and full of lots of great information about the limits of what we can know, and what science is, and what mathematics is.
271
If you love infinity and math, I cannot more highly recommend Matt Parker's Things to Make and Do in the Fourth Dimension.
272
He is hilarious and this book is very very great at explaining some pretty awesome things.
273
So, Keep reading and if you're looking for something to watch, I hope you've already watched Kevin Lieber's film on Field Day.
274
I already did a documentary about Whittier, Alaska over there.
275
Kevin's got a great short film about putting things out on the internet and having people react to them.
276
There's a rumor that Jake Roper might be doing something on Field Day soon, so check out mine, check out Kevin's,
277
and subscribe to Field Day for upcoming Jake Roper action, yeah?
278
He's actually in this room right now.
279
Say hi, Jake.
280
Hi.
281
Thanks for filming this by the way.
282
Guys, I really appreciate
283
who you all are, and as always, thanks for watching.
📺 Mismo canal
B21
What's Inside My Spherical Die?
American AccentUniversity Students
B21
A Cosmic Coincidence
American AccentUniversity Students
B14
My Dad Ranking
American AccentTeensDaily ConversationHigh School Students
C1Is Anything Real?
American AccentUniversity StudentsWorking Adults
B21
Why Are Things Creepy?
American AccentDaily ConversationUniversity StudentsWorking Adults
B24
I Built An Emordnilap Machine
American AccentUniversity Students
C17
The Future Of Reasoning
American AccentIELTS PrepTOEFL PrepUniversity Students
C15
Why Are Things Cute?
American AccentUniversity StudentsWorking Adults
C1All The Ghosts You Will Be
American AccentIELTS PrepTOEFL PrepUniversity Students
✨ Recomendar Video
C14
10 oraciones
The inscribed square problem
American AccentUniversity StudentsWorking Adults
B23
73 oraciones
Box of Goodies | The Big Bang Theory
American AccentDaily ConversationUniversity StudentsWorking Adults
C12
55 oraciones
Convergence and Divergence: The Return of Sequences and Series
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C12
48 oraciones
The Paradox of an Infinite Universe
IELTS PrepTOEFL PrepUniversity Students
C15
10 oraciones
Fourier series
American AccentUniversity StudentsWorking Adults
C12
292 oraciones
Why is Russia So DAMN BIG?
American AccentUniversity StudentsWorking Adults
C12
171 oraciones
The Most Important Picture in the History of Science
American AccentIELTS PrepTOEFL PrepUniversity Students
C186 oraciones
The End of Human Mathematics? AI's Recent Breakthrough -- And What's Next
Business EnglishUniversity StudentsWorking Adults
C12
582 oraciones
You've (Likely) Been Playing The Game Of Life Wrong
Business EnglishUniversity StudentsWorking Adults
C16
104 oraciones
How real are the real numbers, really?
IELTS PrepTOEFL PrepUniversity Students
C13
62 oraciones
A mind-blowing explanation of symmetry | Frank Wilczek for Big Think
IELTS PrepTOEFL PrepUniversity Students
Vocabulario y notas de pronunciación para esta lección
Esta lección de conversación de nivel C1 se basa en el vídeo “The Banach–Tarski Paradox”. Las palabras que más se repiten: piece, infinity, rotation, infinite, sequence. Este vídeo tiene 282 frases y 3386 palabras para practicar shadowing. La parte hablada dura 24:03. El hablante mantiene un ritmo constante de unas 141 palabras por minuto, cómodo para el shadowing. El 84 % de las palabras está entre las 3.000 más comunes del inglés; conviene repasar el resto antes de empezar.
Vocabulario clave de este vídeo
Las 15 palabras más avanzadas del vídeo, con su pronunciación y significado:
| Palabra | Pronunciación | Significado |
|---|---|---|
| infinity sustantivo | /ɪnˈfɪnɪti/ | infinidad |
| rotation sustantivo | /ɹoʊˈteɪʃən/ | rotación |
| sphere sustantivo | /sfɪɚ/ | esfera, bola |
| rotate verbo | /ˈɹoʊ.teɪt/ | girar |
| finite adjetivo | /ˈfaɪ.naɪt/ | limitado, finito |
| paradox sustantivo | /ˈpæ.ɹəˌdɒks/ | paradoja |
| countable adjetivo | /ˈkaʊn.tə.bəl/ | contable |
| backtrack verbo | /ˈbæk.tɹæk/ | retroceder |
| circumference sustantivo | /sɚˈkʌmfɹəns/ | circunferencia |
| uncountable adjetivo | /ʌnˈkaʊntəbl̩/ | incontable |
| unending adjetivo | interminable | |
| dictionary sustantivo | /ˈdɪk.ʃəˌnɛ.ɹi/ | diccionario, tumbaburros |
| decompose verbo | /ˌdiːkəmˈpəʊz/ | descomponer |
| rearrange verbo | /ˌɹiːəˈɹeɪndʒ/ | reacomodar, reacotejar |
| subtract verbo | /səbˈtɹækt/ | restar, sustraer |
Phrasal verbs que vas a escuchar
| Palabra | Significado |
|---|---|
| check out verbo | investigar, chequear |
| turn out verbo | resultar |
| caught up verbo | dejarse llevar por, envuelto |
| come up with verbo | ingeniar, ingeniarse |
| cover up verbo | encubrir, cohonestar |
| cut out verbo | dejar de, dejar |
| cut up verbo | picar, trinchar |
| go through verbo | atravesar |
Frases que vale la pena repetir
Frases cortas y completas del vídeo que puedes reutilizar en la conversación diaria:
- Maybe you've seen it before.
- I mean, where do you even start?
- Banoktarski hasn't left our sights yet.
- Infinity tells us it doesn't matter.
- Could we decompose pieces of it into the whole thing?
Gramática en este vídeo
Las estructuras que más usa el hablante, con las palabras exactas del vídeo:
| Estructura | En el vídeo |
|---|---|
| “Used to” used to + verbo — un hábito o estado pasado que ya no es cierto | used to be |
| Present perfect have/has + participio pasado — una acción pasada que sigue importando ahora | you've seen · we've defined · hasn't left |
| Voz pasiva be + participio pasado — importa lo que ocurre, no quién lo hace | can't be listed · can be paired · can be matched |
Pronunciación a tener en cuenta
El hablante usa 31 contracciones y formas reducidas, como we've, we're, can't. Dilas en su forma corta, tal como las oyes.
- Los sonidos de “th”: theorem /ˈθiərəm/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/
- Los sonidos de “sh” y “zh”: rotation /ɹoʊˈteɪʃən/, dictionary /ˈdɪk.ʃəˌnɛ.ɹi/, illusion /ɪˈl(j)uːʒ(ə)n/, mathematician /ˌmæθ.(ə.)məˈtɪʃ.ən/, ramification /ɹæmɪfɪˈkeɪʃən/
- Palabras largas — cuida el acento: infinity /ɪnˈfɪnɪti/, dictionary /ˈdɪk.ʃəˌnɛ.ɹi/, correspondence /ˌkoɹ.ɪˈspɑn.dəns/, counterintuitive /ˌkaʊntəɹɪnˈtuɪtɪv/, diagonal /daɪ̯ˈæɡ(ə)nəl/
Sonidos difíciles para hispanohablantes:
- s + consonante al inicio — sin añadir una “e” delante: sphere /sfɪɚ/, sliver /ˈslɪvɚ/
- /v/ — no es /b/: los dientes tocan el labio inferior: observe /əbˈzɜːv/, counterintuitive /ˌkaʊntəɹɪnˈtuɪtɪv/, sliver /ˈslɪvɚ/, invent /ɪnˈvɛnt/
- /z/ — sonora, no /s/: decompose /ˌdiːkəmˈpəʊz/, observe /əbˈzɜːv/, clockwise /ˈklɒkwaɪz/, compose /kəmˈpəʊz/, surprisingly /səˈpɹaɪzɪŋli/
Cómo practicar con este vídeo
- Escucha el vídeo entero una vez sin hablar y anota las palabras que no conoces.
- Haz shadowing frase por frase a velocidad normal, repitiendo cada una hasta que tu ritmo coincida con el del hablante.
- Grábate y compara con el original, prestando atención a palabras como infinity, rotation, sphere.
¿Qué es la Técnica de Shadowing?
Shadowing es una técnica de aprendizaje de idiomas respaldada por la ciencia, desarrollada originalmente para la formación de intérpretes profesionales y popularizada por el políglota Dr. Alexander Arguelles. El método es simple pero poderoso: escuchas audio en inglés nativo y lo repites en voz alta de inmediato, como una sombra que sigue al hablante con solo 1-2 segundos de retraso. A diferencia de la escucha pasiva o los ejercicios de gramática, el shadowing obliga a tu cerebro y músculos de la boca a procesar y reproducir simultáneamente patrones de habla reales. Las investigaciones muestran que mejora significativamente la precisión de la pronunciación, la entonación, el ritmo, el habla conectada, la comprensión auditiva y la fluidez al hablar, convirtiéndola en una de las metodologías más efectivas para la preparación del IELTS Speaking y la comunicación en inglés en el mundo real.