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