The Most Controversial Idea In Math

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

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 1.001

then 1.001 and so

on so really what number comes after one

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

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

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

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

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 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

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

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 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