Dangerous Knowledge Page #3

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was a vast new mathematics

of the infinite.

You really finally feel

for the first time,

that the infinite is no longer

this amorphous concept:

well, it's infinite.

And that's all you

can say about it.

But Cantor says:

no!

There's a way you can

make this very precise

and i can make it

very definite as well.

By 1872, Cantor is a man inspired.

He's already grasped and understood,

the nature of real infinity,

which no one before him had done,

but in that same year,

he come's up here to the Alps...

to meet the only other man

who really understood his work:

a mathematician called,

Richard Dedekind.

And this time,

is probably the happiest and most

inspired period of Cantor's life.

Within a year of there meeting, he

announces an astonishing discovery:

that beyond infinity,

there's another larger infinity,

and possibly even a whole

hierarchy of different infinities.

Though it is contrary

to every intuition,

Cantor began to see that some

infinities are bigger than others.

He already knew that when

you looked at the number line,

it divided up,

into an infinite number

of whole numbers and fractions.

But Cantor found that as he

looked closer at this line,

that infinite though the

fractions are, each one...

is separated from the next by

a wilderness of other numbers.

Irrational numbers like pi.

Which require an infinite

number of decimal places

just to define them.

Against all logic,

the infinity of these numbers,

was unmeasurably, uncountably

larger than the first.

What had frightened Galileo,

Cantor had proved:

there was a larger infinity!

Today, Cantor's genius

continues to inspire the work

of some of the

greatest mathematicians.

Greg Chaiton, is recognised

as one of the most brilliant.

Well, infinity was

always there but it...

they tried to contain it.

They tried to...

to keep it in a cage.

And, people would talk

about potential infinity

as opposed to actual infinity.

But Cantor just goes all the way.

He just goes totally berserk.

And then you find that

you have infinities and

bigger infinities and

even bigger infinities

and for any infinite

series of infinities,

there are infinities that are

bigger than all of them.

And you get numbers so big

that you wonder

how you could even name them?

You know infinities so big that

you can't even give them names?

This is just...

It's just fantastic stuff!

So in a way what he's saying is,

giving any set of concepts,

i'm going to invent

something that's bigger.

So this is...

this is paradoxical essentially.

So there's something inherently

ungraspable, that escapes you

from this conception.

So it's absolutely breathtaking.

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