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Veritasium
The Most Controversial Idea In Math
The Most Controversial Idea In Math
Veritasium
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33:00 · Apr 2, 2025
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there
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a
rule
in
mathematics
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there is a rule in mathematics
0:02
that is so simple you would think it obviously
0:05
must be true but if you accept it you find there are now some
0:08
line segments that have no length
0:11
a sphere without adding anything to it can be turned into two
0:15
identical spheres 100 plus years of mathematics
0:18
has been built on this Axiom
0:21
it seems intuitive and it works
0:23
but it also creates
0:24
ridiculous paradoxes so is
0:28
right well it all starts with the issue of
0:31
choice try this choose a number
0:35
I can just pluck a random number from my head like 37
0:38
or 42 but that is the human brain at work not a mathematical
0:43
process in math you can't
0:46
truly pick things at random
0:48
because formulas always give the same result
0:51
which is why computers
0:52
don't have true random number generators
0:54
instead they usually run an algorithm
0:56
on your current local time to generate numbers that appear
1:01
random so if we can't pick randomly
1:04
how do we select anything in math
1:07
well the only way is to follow a rule of some sort
1:10
so a rule could be
1:12
always choose the smallest thing
1:14
for example if we're looking at whole positive integers
1:17
the smallest is one
1:18
for prime numbers it would be two
1:21
easy but what about the real numbers
1:24
that's any number positive negative whole fraction
1:27
even irrational like Pi or the Square < t of two
1:30
now try to choose the smallest one
1:33
it's impossible the real numbers stretch off to negative
1:37
infinity even if we try to fix our rule by making it super specific
1:41
like choose the smallest number after one
1:44
we still get stuck
1:46
there's 1.01 and 1.001
1:50
then 1.001 and so
1:54
on so really what number comes after one
2:00
if we can't begin to specify the order of the real numbers next in
2:04
previous first and last
2:05
we're stuck the ridiculous
2:08
part is we know we have infinite
2:09
options but despite that we can't figure out how to just pick
2:15
one the mission to resolve this began with one man in
2:19
1870 he took on the task of putting the real numbers in a definitive
2:23
order even if it killed him
2:26
and it nearly did
2:31
gorg Cantor was a talented German mathematician
2:34
who found himself at the center of a firestorm
2:37
after publishing one of his very first papers at the age
2:41
29 for centuries our
2:43
understanding of infinity was heavily influenced
2:46
by Galileo's 1638 book
2:48
it raised a key question
2:50
are there more natural numbers
2:52
or are there more square
2:54
numbers just looking at them the square numbers are more spaced out
2:58
and they only become more sparse
3:00
the higher you go
3:01
so it would appear there are fewer squares
3:04
than natural numbers but Galileo
3:06
realized he could draw a line matching
3:09
every natural number with its own square
3:12
and since he could make this one toone mapping
3:14
then he knew that the two sets
3:16
must be exactly the same
3:18
size so there are actually just as many square numbers
3:22
as there are natural
3:24
numbers from this counterintuitive
3:27
result Galileo concluded that terms like more than or less than don't apply to
3:32
Infinity how we normally use them
3:35
it's all just one big concept of
3:37
foreverness and this view prevailed for centuries
3:41
in fact it's how many people still
3:43
understand Infinity today but 200 100 years on
3:47
Cantor wasn't satisfied in 1874
3:51
he wondered what if there were two
3:53
infinite sets out there
3:55
that didn't map perfectly to each other
3:57
would they be different
3:59
infinity so he set out to compare the natural numbers
4:03
and the real numbers between 0
4:05
one caner started by assuming he could
4:08
perfectly map these sets to each other one to one
4:11
so he imagined writing down an infinite
4:14
list with a natural number on one side and a real number between zero
4:18
and one on the other
4:19
since there is no
4:21
smallest real number he would just write them down in any
4:24
order assuming he now has a complete
4:27
infinite list caner writes down another
4:30
real number and to do it he takes the first digit of the first
4:34
number and adds one
4:36
then the second digit of the second number
4:38
and again he adds one
4:40
he keeps doing this all the way down the list
4:43
if the digit is an eight or a nine
4:45
he subtracts one instead of adding to avoid duplicates
4:49
and by the end of this process
4:51
he has written down a real number between Zer and one
4:55
but that number doesn't appear
4:57
anywhere in his list
5:00
it's different from the first number in the first decimal place different from the
5:03
second number in the second decimal place
5:05
and so on Down the Line
5:07
it has to be different from every number on the list by at least
5:10
one digit the digit on the diagonal
5:14
that's why this is called caner
5:15
diagonalization proof and it shows there must be more real numbers between zero and
5:21
one than there are natural numbers extending
5:23
out to Infinity caner
5:27
had revealed something remarkable
5:29
in Infinity doesn't come in just one size
5:32
some infinities like the set of square numbers
5:35
integers or rational numbers
5:37
can be paired perfectly
5:38
with the natural numbers
5:40
you can literally count them 1 2 3 and so on so
5:44
caner called these countable
5:46
Infinities but then there are bigger Infinities
5:49
caner called them uncountable
5:52
these Infinities like the set of all real numbers
5:54
the complex numbers they can't be matched one to one with the natural
6:00
Canter's results rocked the mathematical
6:03
Community after all how can something that continues
6:06
forever be bigger than something else
6:09
that continues forever his work was labeled a horror
6:13
and a grave disease
6:15
but caner wasn't discouraged
6:17
his success only spurred him to pursue his even grander goal
6:20
to show that even uncountably
6:22
infinite sets could be placed in a definitive
6:24
order what caner called a well
6:28
order for a set to be well ordered he required
6:31
two conditions first the set must have a clear starting point
6:36
and second every subset
6:38
a collection of items from that set
6:40
must also have a clear starting
6:43
point so for example the natural numbers are well ordered
6:47
there's a starting point one
6:49
and any subset say 678
6:51
also has a clear starting point in this case six
6:55
you always know which number comes before
6:57
and which comes next
7:01
but what about the integers
7:02
integers stretch off to Infinity
7:04
in both the positive
7:05
and negative directions well Kanto
7:08
realized he could just pick zero
7:10
as the starting point
7:12
and from there his ordering went
7:14
1ga - 1 2
7:16
-2 ranking the integers
7:18
by their absolute value
7:19
their distance from zero
7:21
it doesn't matter if you put the positives
7:23
first or the negatives first as long as you're consistent
7:26
ordering them this way is actually what allows us to Max the integers to
7:30
the natural numbers and see that both sets are the same size
7:34
but there are other ways we could well order the integers
7:37
we could start with zero
7:39
and then have 1 2 3 all the way to positive infinity
7:42
and then - 1
7:44
-2 -3 all the way to negative
7:47
Infinity this is not how we're used to counting
7:50
but both of these options fit the definition
7:52
of a well ordering
7:54
there's a clear starting point zero
7:56
and all their subsets
7:58
also have a definitive
8:00
point caner had successfully
8:03
well-ordered a set that was infinite in both directions
8:07
but it was only countably
8:08
infinite in his next book he published his well-ordering
8:12
theorem it claimed that every set
8:15
even the uncountably infinite ones like the real numbers
8:18
could be well ordered
8:21
the problem was he hadn't actually proven this
8:24
because he couldn't every method he tried
8:28
had failed but there was one big reason that caner was so confident in
8:33
his theorem caner was a devout Lutheran
8:36
and he believed God was speaking
8:39
through him he said
8:41
my theory stands as firm as a rock
8:44
every arrow directed against it will return quickly to its Archer
8:48
how do I know this
8:49
because I have studied it from all sides for many years
8:53
and above all because I have followed its roots so to speak to the
8:57
first infallible cause of all creation ated
9:00
things belief not withstanding
9:03
the well-ordering theorem was a lofty claim to make without any mathematical
9:07
proof and so for the second time
9:11
the mathematical Community attacked
9:13
and caner leading the charge was Leopold
9:17
chroniker the head of mathematics at the University of Berlin
9:21
chroniker completely dismissed Canter's work
9:23
labeling him a scientific charlatan
9:26
and a corruptor of the
9:27
youth and chroner used to be Canter's
9:30
teacher caner dreamed of joining him at the University of Berlin
9:35
but all his applications
9:36
were mysteriously denied so caner took the rejection
9:40
personally in 1884 he wrote 52
9:44
letters to a friend
9:45
and every one of them
9:47
bemoaned chroniker soon caner suffered what would be the first of many nervous breakdowns
9:53
he was confined to a sanatorium
9:55
for Recovery the only way he could prove every when wrong
10:00
was by well ordering the real numbers
10:03
but he couldn't find a starting point
10:06
literally once caner was released from the sanatorium
10:09
he stepped away from math a Broken Man
10:12
and over the next 15 years he taught philosophy
10:15
and rarely dabbled in his old
10:17
Pursuits perhaps his greatest challenge came at the 1904
10:21
International Congress of mathematicians
10:23
there Julius kunig a respected Professor from Budapest
10:27
announced he had proof that caner
10:29
well-ordering theorem was wrong
10:32
in the audience was not only Cantor
10:34
but also his wife
10:36
two of his daughters
10:37
and his colleagues he felt
10:40
utterly humiliated but there was also
10:43
another in attendance Ernst
10:46
zero zero was a German mathematician
10:49
who had recently developed a keen interest in Canter's
10:51
work and as he listened to kun's presentation
10:55
something felt off within 24 hours zero
10:59
had pinpointed the problem
11:00
kik's proof contained a damning
11:03
contradiction and within a month zero
11:06
published a three-page article titled
11:08
proof that every set
11:10
can be well ordered
11:12
and it was Flawless
11:15
Zero's breakthrough came when he discovered something profound
11:18
in Canter's work a mechanism
11:20
which caner uses unconsciously
11:22
and instinctively everywhere but formulates
11:25
explicitly nowhere see all along
11:29
aner had been assuming
11:30
that he could make an infinite number of choices
11:32
at once from any set including
11:35
uncountably infinite sets like the real numbers
11:38
but this was just an assumption
11:40
nowhere in the mathematical
11:42
rule book was this explicitly
11:43
permitted and math is built on rules
11:46
specifically axioms axioms are simple statements
11:49
we accept as true without proof
11:52
zero realized Canter's assumption needed to be formalized
11:55
into something that holds up in a system of proof
11:59
a new Axiom that said
12:01
making all of those choices
12:02
was possible he needed
12:04
the Axiom of choice
12:07
the axum of choice can be said in the sense that if you have
12:10
infinitely many sets and each set is not empty then there is a way
12:13
to choose one element from each of the sets
12:16
for finite sets this seems obvious
12:19
just go setby set and pick something
12:22
even for infinite sets
12:23
it's easy if there's a clear rule
12:26
like always choose the smallest thing
12:28
but sometimes there is no natural rule
12:31
in those cases when you're choosing from infinitely many sets
12:35
including the uncountable ones you need the axium of choice
12:39
we can't say how we're choosing
12:40
but the Axiom makes all of these choices
12:43
all at once the Axiom
12:45
doesn't allow you to say which element you've chosen
12:48
only that infinitely many choices
12:51
are possible so how does this new Axiom
12:54
enable us to well order the real
12:57
numbers zero uses the axium of choice to choose a number from the set
13:01
of all real numbers
13:03
he places this number let's call it X1
13:05
into a new set R
13:07
the Axiom then allows him to choose another number from the subset of all
13:11
reals minus the one taken out
13:13
he calls this number X2
13:15
and places it as the next number in his set
13:18
and he keeps doing this taking the chosen number and placing it next X3
13:22
X4 X5 now it feels like he's choosing these numbers one at a time
13:26
but in reality the choices are made
13:29
from all possible subsets
13:31
at the same time
13:32
as zero indexes each number with the natural numbers
13:36
at first it might seem like he'd run into a problem
13:38
because the natural numbers are only countably
13:41
infinite whereas there are way more reals
13:44
so he should eventually run out of labels
13:47
but we can count beyond
13:49
Infinity we did it earlier when we counted past positive Infinity to get to
13:52
neg1 -2 and so
13:55
on so we just need a new set of numbers
13:58
that extends past the naturals
14:00
call the next number
14:01
Omega then Omega + 1 Omega + 2 and so on
14:06
these Omega numbers are not bigger than infinity
14:09
they just come after
14:10
infinity they don't tell us how many things are there but they do tell
14:14
us their order so the next number we pull out we'll label it X
14:18
Omega then X Omega + 1 x Omega plus 2 and so on
14:23
this will continue until we match the size of the real numbers
14:27
and our original set is
14:29
now every real number is in our new set
14:32
there is a first number X1
14:35
and every subset also has a first number
14:38
and just like that
14:39
we have successfully well-ordered
14:41
the real numbers this order looks nothing like our familiar ordering
14:46
a billion could come before 02
14:49
but with this process
14:50
we can prove that a well
14:53
exists and more than that we now have a way to resolve our issue
14:57
of how to choose mathematically
14:59
we can't pick a smallest
15:00
real number but now we can pick a first real number our starting point
15:05
and we can do this for any set meaning
15:08
all sets can be well ordered
15:09
no matter the infinity
15:11
so Canter's well-ordering theorem and Zero's
15:14
axium of choice are
15:18
equivalent caner was so
15:20
relieved zero had proved the well ordering theorem and well ordered the real numbers
15:26
all in under a month
15:30
zero took something mathematicians
15:32
had unknowingly relied on for decades
15:34
and turned it into a formal Axiom
15:36
he showed that understanding math isn't just about numbers
15:40
it's about the logic behind them
15:42
and lately I've been trying to do a similar thing but with AI
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where I'm trying to understand the logic
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this video and now back to the Axiom of
16:56
choice the Axiom of choice may have been a new idea
17:00
but its use was anything but
17:02
zero scanned dozens of papers from other mathematicians
17:05
and realized they had also been using the Axiom all along
17:09
even those who had criticized
17:10
Canter's work it just goes to show
17:13
how unintuitive it is that it's even an axiom
17:16
people had been using it
17:18
for like a decade
17:20
un unknowingly but this almost seems too obvious
17:24
Zero's proof didn't actually construct
17:26
a well order it just said one must
17:29
exist but can something exist if we can't actually build it
17:33
his proof also used an uncountable
17:36
number of steps was that even allowed
17:39
some mathematicians argued proofs should be finite
17:42
others accepted Infinity but only the countable
17:45
kind and then things got
17:48
worse when mathematicians played around with the Axiom of choice
17:52
it created disturbing results
17:55
one of the first came from jeppi
17:57
Vitali in 1905 Vitali
18:00
used the axium of choice to build a set of numbers that shattered
18:03
our idea of what it means for something to have
18:06
length so what Vitali
18:08
does is he takes
18:09
every real number between zero and one
18:12
and assigns it to one
18:14
of an infinite number of bins
18:17
let's call these bins
18:18
groups so we want each real number to end up in exactly
18:23
one of our infinite bins
18:26
so how does he do it
18:27
well let's say we have two numbers
18:30
X and Y if their difference
18:34
x - Y is equal to a rational
18:37
that is one integer
18:39
divided by another integer
18:41
well then both X and Y will go into
18:45
the same bin but
18:48
if uh we have two other numbers
18:50
let's say p and
18:52
Q and their difference
18:54
is not irrational so it's an irrational
18:57
difference with then those two numbers
19:00
will go into separate
19:03
bins so let's do some examples
19:06
if this is 3/4
19:08
minus a half then we get a quar
19:11
and so both 3/4
19:13
and a half will go
19:14
into the same bin
19:17
in fact you can see that all rational
19:19
numbers from this span 0 to 1 they'll all end up in the same
19:24
group now if you have irrational
19:26
numbers well it's not clear
19:28
whether they will go into the same bin or not because for example
19:32
if we have the number
19:33
< tk2 over2 minus
19:36
say < tk2 over 2
19:38
- a/4 well then that does have a rational difference even though each of
19:42
these numbers is irrational
19:44
so these two numbers
19:46
will go into the same
19:48
group but if we have a rational number is < tk2
19:52
over 2 - < tk2
19:54
over 3 well that gives an irrational
19:57
difference so < tk2 2 over 3 will have to go
20:00
into a different bin
20:02
and it will be joined by all of the numbers it has a rational
20:06
difference from and in this way
20:08
you can assign each real number to
20:11
exactly one of these bins
20:13
next Vali used the axium of choice to reach into
20:18
each group and select
20:19
exactly one number which would be a representative
20:23
of the group so we could pull out 3/4
20:25
from the rational group
20:27
< tk2 over2 from this group two over three from that group and so
20:30
on though of course
20:31
because we're using the Axiom of choice you don't actually know
20:35
what that representative number is
20:37
just that you have one
20:39
so we could write it
20:40
down like this we have these representatives
20:44
from each group and together they form
20:47
the Vitali set you can visualize
20:49
this set as a collection of points between
20:52
Z and one next
20:54
Vitali makes infinite copies
20:56
of his set and each one he shifts
20:59
by a different rational
21:01
number between - 1
21:03
and positive 1 so if you think about what that does
21:06
it's going to move each representative
21:08
number to be at the position
21:10
of every other number in its group
21:13
if we just had the one
21:15
rational number that we plucked out as a representative
21:17
from the rational group now we're going to shift it by every
21:20
possible rational number between
21:22
1 and positive one so it's going to end up at every other position
21:26
occupied by the other members of its group
21:28
at least on the span between
21:30
zero and one so if you imagine
21:33
now merging all of these infinite sets together
21:36
there's going to be no overlap
21:38
between the points and second
21:40
we are going to have every real number between
21:43
0 and one because
21:45
on that span we have
21:47
every member of every group
21:49
so now the question is
21:51
what is the size
21:53
of the vital set
21:54
now we know that the union of those sets
21:57
must be greater than or equal to
22:00
one because we have every real number between
22:03
0 and 1 but also these points only extend out as far as
22:08
-1 or pos2 so it must be
22:12
less than or equal to
22:14
3 but this is where the problem arises
22:17
because what number for the size of the vital set could you add to
22:22
itself infinitely many times
22:24
and end up with a value between 1 and three
22:27
there is no number like that
22:29
I mean if the size of the Vitality set was Zero
22:32
you add it up infinitely many times
22:34
you still get zero
22:36
if the size of the Vitality set is a small positive value
22:39
then you add it up infinitely many times you're going to get infinity
22:42
not three so we have a contradiction
22:45
and the only way out
22:46
is if the Vitality
22:48
set itself is unmeasurable
22:50
which seems crazy non-measurable
22:54
sets like the vital set have no consistent
22:57
definition of size or length or area
23:00
or even probability but math is built on the idea that everything can be
23:05
Quantified whether it's distance
23:06
time or weight except
23:09
now there are non-measurable
23:10
sets and it seems like the Axiom of choice
23:13
is blame this was just the start of the Uproar caused by the Axiom
23:19
in 1924 two mathematicians
23:21
Stephan banck and Alfred tarsky
23:23
used it to show
23:24
something that looks like a magic trick
23:27
they proved you could take a single solid ball and split it into just
23:31
five pieces and then by carefully
23:33
rotating and moving those pieces
23:35
you could reassemble them into two balls
23:38
each identical to the one we started with
23:41
and you could keep going
23:42
until eventually you have an infinite
23:44
number of balls Infinity
23:47
all from one this sounds absurd
23:51
but we can actually see how it works by building a
23:55
graph imagine you can move in four directions
23:58
up down left and right
24:00
after taking a step say to the left
24:03
you get the same four choices
24:05
up down left and right
24:07
but if you go to the right you'll end up back where you started
24:10
so the Only Rule we're going to have is that you can't immediately
24:13
reverse a move and we'll keep repeating this at every step
24:17
drawing each new line
24:19
half the size of the previous one so it all fits on the screen
24:23
if we keep going we'll end up with this infinitely
24:26
branching graph looking at our graph we can break it into five sections
24:32
there's the middle section where we started
24:34
and then there are four other sections
24:36
that are all identical
24:37
just rotated so if we take this section to the left and we move
24:42
everything one step to the right
24:44
the top part ends up here the bottom part here and the leftmost part
24:48
here then we've almost recreated
24:51
the entire graph the only thing we're missing
24:54
is this section so let's add it back in
24:57
but we could have done the same same thing in a completely
24:59
different way by taking the bottom section and moving it one step up
25:04
now the leftmost part ends up here the rightmost part here
25:08
and the bottom here
25:09
again we're just missing one section so let's add it back in
25:13
but this means I can recreate
25:14
the entire original graph in two completely
25:17
different ways we took one graph
25:20
split it into sections
25:21
shifted the sections so the left section went to the right and the down
25:24
section up and somehow
25:26
ended up with to
25:28
identical copies this is exactly what benck and tarski did but with a ball
25:34
like our graph we again have four moves
25:37
we can rotate the ball
25:38
up down left or right
25:41
and again our only rule is that we can't immediately
25:44
reverse a move and to make sure we never come back to the same
25:47
point every rotation will be by the same
25:50
irrational portion of a circle
25:52
we can pick a random starting point
25:54
mark it and then start rotating the ball
25:58
each point is colored
26:00
based on the direction of rotation
26:02
used to get there
26:04
if we do this an infinite number of times
26:06
we end up with this collection of points
26:09
this is a countably
26:11
infinite collection because we could list each rotation
26:14
and assign it a natural
26:15
number but the surface of a ball has uncountably
26:18
infinite points just like the real number line
26:21
so if we want to cover the entire
26:23
surface we would need to repeat
26:26
this process but where do we start
26:28
next since there are uncountably
26:30
infinite possible starting points we can't list them all
26:34
and we want to be sure to avoid any points we've already colored
26:38
so the solution is to use the Axiom of choice
26:41
with it we can keep choosing unique starting points
26:45
even though we can't say
26:46
exactly how we are choosing
26:48
them once we've colored every point on the ball
26:51
we can split the points into five groups
26:54
one for the starting points
26:55
and four others based on the final rotation
26:58
used to arrive at those points
27:00
these groups can now be treated
27:02
just like the sections of our graph
27:04
we can take the group of points that end with a left rotation
27:07
and rotate it to the right
27:09
then we add in the group that ends with a right rotation
27:12
and just like that
27:13
we've recreated our original
27:15
ball and we can do it again
27:18
making an extra move to account for the starting points
27:20
we can equally take the group that ends with a down rotation
27:24
and rotate it upwards
27:25
then we add in the group that ends with an up rotation
27:28
and our starting points
27:30
and now we've recreated
27:31
our original ball a second
27:34
time now this is a bit of an oversimplification
27:37
but it gives you the essence
27:38
of how this is done
27:40
from one ball we have created two identical balls of the same volume
27:45
and Nothing Stops us from doing this again two balls can become four four
27:48
become eight and before you know it you've got infinite
27:52
balls the axium of choice is something that's so obviously
27:55
true and its consequences
27:57
are so obviously false
28:00
that you're like what the hell is going
28:02
on this infinite duplication
28:04
is theoretically possible but the catch is the groups we split the ball into
28:09
aren't simple shapes they're actually non-measurable
28:12
just like the vital
28:13
set although the original ball has a volume and the duplicated
28:16
balls have a volume
28:18
the step in between
28:19
violates our understanding of size
28:22
this is what allows
28:23
the Paradox to happen
28:26
of course those are not physically
28:28
plausible Cuts but like there's a more
28:31
uh metaphysical question like should this even remotely be possible if we could make
28:35
such cuts and the answer to almost every human I know is absolutely
28:39
not the truth is
28:41
no one knew what was going on
28:43
that same year tarski tried to push the Axiom of choice further
28:47
proving it is equivalent
28:48
to the statement that squaring
28:50
any infinite set would not increase
28:52
its size when tarski first submitted this work to a journal in Paris
28:57
the editor leag responded
28:59
dismissively nobody's interested in the equivalence
29:02
between two false statements
29:05
not to be deterred
29:06
tari sent it to a different editor at the same Journal
29:09
forche his response nobody's interested in the equivalence
29:13
of two obviously true
29:15
statements tarski never submitted a paper there
29:19
again so math was in crisis for over 30 years
29:24
with people not knowing
29:25
what to believe the question is
29:28
wait a second is this really an axiom
29:30
or is this something that you can prove
29:32
in 1938 we finally started getting some
29:36
ansers the Austrian mathematician
29:38
Kurt goodle proved there is a world where all the other already accepted axioms
29:43
of set theory hold true
29:45
and so does the Axiom
29:46
of choice then in
29:48
1963 Paul Cohen proved there's also a world where all the axioms of set
29:53
theory hold true except
29:55
for the Axiom of
29:56
choice this is kind of like the parallel postulate
30:00
in Geometry you can think of geometry
30:02
as a game the first four postulates
30:05
or axioms are like the minimum rules
30:07
required to play that game
30:09
and then the fifth Axiom
30:10
selects the universe that you want to play in
30:13
if you choose that the fifth Axiom doesn't hold so there are no parallel
30:17
lines then you're playing in spherical
30:19
geometry if you choose one parallel line you're playing in flat geometry
30:24
and if you choose more than one parallel line
30:27
then you're playing in hyperbolic
30:28
geometry all of these geometries
30:31
are valid it just depends on the math you want to
30:35
do and it's the same for the Axiom of choice
30:38
the Axiom of choice can neither be proven nor disproven
30:41
from the other axioms
30:43
so as long as the other axioms are consistent
30:46
adding Choice won't lead to any contradictions
30:49
Paul Cohen was Award of the fields medal 3 years later for his groundbreaking
30:53
result as well as his other work in set
30:56
theory and after good and Cohen's work
30:59
most of the debates about the Axiom of choice died
31:01
down in the end what the hell is going on is that it's up
31:05
to you whether you want to choose
31:07
for the aim of choice to be
31:09
uh a part of your system or not
31:11
and face the consequences
31:13
of either having it or not having it
31:15
despite the counterintuitive results created by the axium of choice
31:18
like non-measurable sets and infinite
31:21
duplication it is incredibly
31:23
useful Choice allows mathematicians
31:25
to replace lengthy explicit proofs
31:28
with more concise arguments
31:30
by proving statements in the finite case many proofs can be extended to any
31:34
infinite case in just one line
31:37
This reduces proofs that could have been 20 pages to just half a page
31:42
and the Axiom of choice doesn't just make math easier
31:45
it is essential to some proofs
31:47
there are many theorems where the general case can't be proven
31:50
without using Choice somewhere
31:53
now some mathematicians still prefer proofs without Choice
31:56
even if it's harder
31:57
the proof has to be spelled out step by step to
32:00
generalize to infinite cases and this provides additional information
32:05
some mathematicians spend their time studying universes
32:07
without the Axiom of choice
32:09
to understand what happens when we remove it
32:12
but today the axium of choice is almost universally
32:16
accepted for the past 80 plus years
32:19
generations of mathematicians have been taught
32:21
with Choice as a given to the point where
32:24
many who use the axium of choice might not even realize when they're doing
32:28
it if you don't include the axium of choice then you're kind of working
32:30
with both hands tied behind your back
32:32
it's very hard to make any progress
32:34
on Modern math so the question
32:37
was never really is the Axiom of choice right
32:40
but rather is the Axiom of choice right
32:43
for what you want to do
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