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Forums > Social Chat > Is the internet infinitly big?

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caz86


Twister Fire Starter ridin on a tratter
Location: Bristol/Exeter UK

Total posts: 156
Posted:Is the internet infinitly big?

We were discussing this in Maths today. I'm interested in your views!


Caz

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:sort of.

its a different context but yes, in the hypothetical, this would be 1% of that infinity.

a good example of this type of idea of infinity in the real world is a ring of 100 juggling balls with 1 of them being orange with blue spots (the other 99 being red lets say).
we are then allowed to walk around this ring but have no outside references or indication of where on it we are.
if we then count the number of orange balls we pass against the red balls we tend towards a 1:99 ratio which this gives us an idea of the overall proportions.
but we cannot do anything with this proportion unless we have an identically proportioned infinity elsewhere.

to realise why this fraction is meaningless outside of your hypothetical infinite juggling ball factory, consider taking all the orange balls away.
now you have an infinite amount of orange balls with blue spots and an infinite amount of red balls.

unless we are in the unique situation that we know of an identical infinity to exchange with, for all other incidences the orange ball infinity and the red ball inifinty are equal.


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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Matthew B-M


Matthew B-M

Lemon-Aware Devilstick-wielding Operative
Location: East London Wilds

Total posts: 605
Posted:Dom, and of course, you're forgetting the countable and uncountable infinities.

The only useful fraction of infinity is 1/infinity of infinity, which may or may not be finite (see above about countable and uncountable), if it is finite, it's more useful to think of limits as things tend to infinity, rather than what happens at the infinity itself.

Why is it that I still remember this stuff? I haven't used it in 4 years...


Luv 'n' Lemons
purity :: clarity :: balance

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Matthew B-M


Matthew B-M

Lemon-Aware Devilstick-wielding Operative
Location: East London Wilds

Total posts: 605
Posted:Coleman: if there really is an infinity there, they are equal, because they're both countable infinities. And of course, not only that, but there'd be the same number of balls as there were atoms in said juggling balls, because similarly, it is a countable infinity. There is one countable infinity and one uncountable one. (also known as aleph-nought, and aleph-one).

Bloody physics degree. It's made me into a complete geek.


Luv 'n' Lemons
purity :: clarity :: balance

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:cardinal numbers and limits of functions eh?

this is a real test of memory innit!

man it could so turn geek-ridden in here - but unless anyone objects, i think the original topic of the thread has been decided so an infinity discussion kinda follows...

for anyone that cares about the difference, i think the example i was given went something like:

aleph-null is the countable infinity.
it is an infinity of discrete, countable numbers: 1, 2, 3, 4, 5...etc.
the designation aleph-null suggests the existence of aleph-one.
but how do we go beyond aleph-null?
suppose we count from one to aleph-null in one hour, and after that increase our speed, so that we counts from one to aleph-null in half an hour, then again in 1/3 of an hour, and continue harmonically decreasing the time it takes to get from one to aleph-null.
we would never reach aleph-one.
aleph-one is the limit towards which one is counting.

hence aleph-one is generally regarded as the complete set of real numbers or 'the continuum' (C).

the thing i remember that confused me was the question 'is there an infinite number between aleph-null and aleph-one' to which maths answers "i don't know" (the undecided continuum hypothesis)
that blew my mind.
you can define a set theory to show there is and you can also define a set theory to show there is not - we will never know.

so what mathew b-m was saying was this:

aleph-null^aleph-null = aleph-null

ie. countable infinities are equal, whereas with alpeh-one (which we will define as 'C'):

C^C > C

and if say x = C^C then X^X is an even greater infinity.

and that is what qft makes use of.

phew! my brain does still work
its a proper headf*ck for a friday afternoon though


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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oli
SILVER Member since Jul 2003

not with cactus
Location: bristol/ southern eastern devo...

Total posts: 2052
Posted:why is it so interesting to think about infinity like this?

Me train running low on soul coal
They push+pull tactics are driving me loco
They shouldn't do that no no no

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Matthew B-M


Matthew B-M

Lemon-Aware Devilstick-wielding Operative
Location: East London Wilds

Total posts: 605
Posted:How else would you think about it?

Luv 'n' Lemons
purity :: clarity :: balance

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Stebbins
BRONZE Member since Dec 2002

10th degree spoon weilder
Location: Halifax, Canada (currently in ...

Total posts: 171
Posted:To be completely honest I really don't think that the human mind can comprehend concepts like this correctly.
Can a bug understand genetic engineering?
We can but aren't we just a more advanced bug in the cosmic spectrum of things? How can we say "this is how it is" if we don't even understand the nature of our own existence? I think that there are much bigger and more advanced bugs out there. We have many good and interesting hypothesis on the go but in the grand scheme of things they probably are all waaaaaaaaaaay of. And that's my 2 cents..


-Beeaaatch please, I'm the macaroni with the cheese.
-This message will self destruct in 10 seconds.

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funkymonky


funkymonky

member
Location: oxford

Total posts: 192
Posted:i was in a conversation yesterday with a friend. the topic was that if alians and/ or other plans of existance existed, why hadn't they told us about themselves and explained what they wanted, if anything.


my answer...

"have you ever felt the need to explane yourself to a cockroach?"

think about it kids......


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oli
SILVER Member since Jul 2003

not with cactus
Location: bristol/ southern eastern devo...

Total posts: 2052
Posted:quote:Originally posted by Matthew B-M:
How else would you think about it? just as a really big number...


Me train running low on soul coal
They push+pull tactics are driving me loco
They shouldn't do that no no no

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DarkDevil


DarkDevil

member
Location: St. John's, Newfoundland.

Total posts: 233
Posted:If you take the total of all hardrives ever produced and multiply it by the average size of a harddrive then you will have the maxiumum size the internet could possibly be at that time.
If you take the total number of homeusers and multiply it by the percent of people who host stuff off of their own computer and add that to the number of computers worldwide subtracted by the number of homeusers and mutliply it by the average harddrive size of... 60... then thats a rough way of estimating the total size of the internet.


Ladies and gentlemen, this is not a trick, this is a stunt. The difference between a stunt and a trick is that a trick is something that tricks or fools the audience, a stunt is something that only an idiot would do -Ivan, Ash Circle, 10/04/03

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Briggy


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Location: england

Total posts: 18
Posted:/boggle

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Matthew B-M


Matthew B-M

Lemon-Aware Devilstick-wielding Operative
Location: East London Wilds

Total posts: 605
Posted:quote:Originally posted by oli:
quote:Originally posted by Matthew B-M:
How else would you think about it? just as a really big number... Right, but you can't, because it's not a "really big number", it's infinitely big. And there's your problem. So you get weird effects. Calculus is all about the limits as things tend to infinity (or to finite points where infinities would occur), and is vital to being able to do large amounts of modern physics. "0/0" is meaningless out of context, but "the limit as x tends to 0 of x/x is going to be different to "the limit as x tends to 0 of sin(x)/x" and so on. Both of those terms are finite at x=0, despite being uncalculable at that point, because you get infinities involved.


Luv 'n' Lemons
purity :: clarity :: balance

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SlightlySinged
GOLD Member since Jun 2003

member
Location: Melbourne, Australia

Total posts: 82
Posted:This is how I think of infinity.

Question: Is the universe of infinite size?

When ever I ask someone this question, they always answer yes immediately. Then I tell them the following:

If the universe is of infinite size, then that means infinite possiblities. Saying there is finite mass but infinite size is silly - how would you ever know there is finite size - you will never get to the end of the universe to find out.

Okay, so if you choose a particular direction, and travel far enough, you will come across a galaxy vaguely similar to our own.

Keep going. You will come across a galaxy VERY similar to our own.

Keep going. You will, by the law of infinity, eventually come across a galaxy that is EXACTLY like our own. With an Earth. With an organism identical to you, reading this on a computer exactly the same as yours.

If you keep going and going, you will come across an INFINITE number of earths all identical to our own. Now that's just one direction. The universe being infinite, you can travel in all infinite number of directions and come across an infinite number of organisms that are identical to you.

Infinity is not just a big number. That's like saying zero is just a really small number.

Our internet is not infinite in size, but there are an infinite number of internets.

(I would love to hear a counter argument for the infinite universe theory - don't be shy to try and shoot me down!!!)


I do poi nearly every day. But it's not like I'm addicted or anything. I mean, sure, I am always conscious of exactly where my poi are at all times, but I'm not obsessed. um.. Anyone have the number for Poi-ers Anon?

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flid
BRONZE Member since Aug 2002

flid

Carpal \'Tunnel
Location: Warwickshire

Total posts: 3136
Posted:hmmm, i wrote a long post for this, but can't be arsed to argue the points. I leave mbm on geek duty for this thread

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SlightlySinged
GOLD Member since Jun 2003

member
Location: Melbourne, Australia

Total posts: 82
Posted:One more thing.

Someone said if you have a ring of balls, and every 10th one is a different colour, and you walk around the ring and infinite number of times, then you will count out a 1/10 fraction of infitity.

I do not believe this is true. You absolutely cannot have a fraction of infinity. You CAN, however, have a RATIO, which when talking about infinity, is a different animal. The ratio of blue to orange is 1:10. As soon as you seperate the blue balls from the orange balls, and say that's 1/10 of infitiy, what you actually have is an infinite number of blue balls. If you then add the blue balls with the orange balls, you won't be able to work out the fraction, since you will simply have an infinite number of blue balls and an infinite number of orange balls. You won't even be able to work out the ratio.

You might try to say that if you lined up the blue balls next to the orange balls, the line of orange balls would be longer, but that's also not true. Both lines would be infinitely long, ie, no end. You wouldn't be able to work out any sort of ratio or fraction.

Halve any number an infinite number of times, and you theoretically get zero. What do you get if you halve infinity an infinite number of times? Zero? Or is it still infinity? It'd have to still be infinity, so infinity is not a number. It is the culmination of all, just as zero is the lack of anything.

Also, I do not believe you can count to infinity within a finite time. If you tell me a time, I'll say so what's the next number after that time? You either need infinite time to count to infinity, or infinite speed of counting, in which case the time it would take to count to infinity is NO TIME AT ALL. Doing anything at infinite speed means you get it done instantly.


I do poi nearly every day. But it's not like I'm addicted or anything. I mean, sure, I am always conscious of exactly where my poi are at all times, but I'm not obsessed. um.. Anyone have the number for Poi-ers Anon?

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simian


simian

110% MONKEY EVERY TIME ALL THE TIME JUST CANT STOP THE MONKEY
Location: London

Total posts: 3149
Posted:quote:Originally posted by SlightlySinged:
(I would love to hear a counter argument for the infinite universe theory - don't be shy to try and shoot me down!!!)Alrighty then

quote:If the universe is of infinite size, then that means infinite possiblities. Saying there is finite mass but infinite size is silly - how would you ever know there is finite size - you will never get to the end of the universe to find out.No. Infinite size does not mean infinite mass. Saying the universe is infinite just means that it doesn't have boundaries. You can go as far as you like in any direction and never hit the brick wall at the end of the universe. It doesn't mean that it's full of an infinite amount of stuff.

And even if there was an infinite amount of stuff, that doesn't mean that it'll exist in every possible configuration (eg. alternate Earths). Just because something is infinite, it doesn't follow that it will exhaust every possibility of its variations. You could have an infinite series of coin flips, with the coin never coming up tails. It's unlikely but possible.

One more thing...

quote:Halve any number an infinite number of times, and you theoretically get zero.No you don't. That's the basis of Xeno's paradox (the one with the Tortoise that reckons it can outrun arrows). You never get zero, you just get really really really small numbers. Zero isn't just a really small number. That's like saying infinity is just a big number

Remember, the sum of an infinite series is not necessarily an infinite number. That knowledge can save your tortoise.

[ 10. November 2003, 23:16: Message edited by: simian ]


"Switching between different kinds of chuu chuu sometimes gives this "urgh wtf?" effect because it's giving people the phi phenomenon."

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Matthew B-M


Matthew B-M

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Posted:Not quite true. The thing about xeno's paradox is that you get closer and closer to a time when the arrow hits. Time doesn't actually work like that, so the arrow hits the tortoise. This means that SlightlySinged is in fact correct.

Anyway, if you believe the Big Bang theory, then the Universe is finite but expanding, because it's defined by the photons leaving from the Big Bang.

This thread is really making me realise how much of a geek I am. flid!


Luv 'n' Lemons
purity :: clarity :: balance

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:quote:Originally posted by SlightlySinged:
Someone said if you have a ring of balls, and every 10th one is a different colour, and you walk around the ring and infinite number of times, then you will count out a 1/10 fraction of infitity.hello, i'm 'someone' i think.
although i didn't say that.
i said:

quote:Originally posted by coleman:
a good example of this type of idea of infinity in the real world is a ring of 100 juggling balls with 1 of them being orange with blue spots (the other 99 being red lets say).
we are then allowed to walk around this ring but have no outside references or indication of where on it we are.
if we then count the number of orange balls we pass against the red balls we tend towards a 1:99. ratio compare and contrast.
important words are 'tend' and 'ratio'.


i also said:

quote:Originally posted by coleman:
to realise why this fraction is meaningless outside of your hypothetical infinite juggling ball factory, consider taking all the orange balls away.
now you have an infinite amount of orange balls with blue spots and an infinite amount of red balls. to which you replied:

quote:Originally posted by SlightlySinged:
You CAN, however, have a RATIO, which when talking about infinity, is a different animal. The ratio of blue to orange is 1:10. As soon as you seperate the blue balls from the orange balls, and say that's 1/10 of infitiy, what you actually have is an infinite number of blue balls. If you then add the blue balls with the orange balls, you won't be able to work out the fraction, since you will simply have an infinite number of blue balls and an infinite number of orange balls. You won't even be able to work out the ratio.is it me or is there an argumentative echo in here


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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simian


simian

110% MONKEY EVERY TIME ALL THE TIME JUST CANT STOP THE MONKEY
Location: London

Total posts: 3149
Posted:quote:Originally posted by Matthew B-M:
Not quite true. The thing about xeno's paradox is that you get closer and closer to a time when the arrow hits. Time doesn't actually work like that, so the arrow hits the tortoise. This means that SlightlySinged is in fact correct.

Divide something infinitely, you will never reach a point where its factor is zero.

The paradox hinges on saying that if you can divide something infinitely, then it must be infinite. And you can divide everything infinitely. So everything is infinite.

Which is rubbish. Because: the sum of an infinite series is not necessarily an infinite number.

So the tortoise is indeed kebab-ed.

BTW: "time doesn't work like that so the arrow hits the tortoise"

Thats the worst explanation of Xeno i've ever heard. i suppose that's what happens when science types get let loose on philosophy


"Switching between different kinds of chuu chuu sometimes gives this "urgh wtf?" effect because it's giving people the phi phenomenon."

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:get your rulers and elastic bands out lads, its a...

GEEK BRAWL!


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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Posted:

OK boys slide rules at the ready...

10 paces...
turn and...

Theorise!

PsyB.


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simian


simian

110% MONKEY EVERY TIME ALL THE TIME JUST CANT STOP THE MONKEY
Location: London

Total posts: 3149
Posted:Ha, you are having no hope of destroying me!!!

I are utilise my Final Poindexter Attack!!!

sorry, came over all badly dubbed for a second...


"Switching between different kinds of chuu chuu sometimes gives this "urgh wtf?" effect because it's giving people the phi phenomenon."

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SlightlySinged
GOLD Member since Jun 2003

member
Location: Melbourne, Australia

Total posts: 82
Posted:hahaha Sorry, Coleman, I didn't read your post very well, obviously

Okay, so if the universe is of finite size, what's one meter further away?

If the universe is of infinite size, then it must be of infinite mass. If there is one galaxy, there must be an infinite number of galaxies, and if there are an infinite number of galaxies, then there must be an infinite number of permutations, hence any possibility must occur. Infinity means anything is possible.

Nice try with the coin toss argument. By the rules of probability, it is possible that if you toss a coin huge number of times, it will come up heads every time. The likelihood is very very small, but it's still likely.

However, I don't believe probability works at infinity. I would argue that if you toss a coin an infinite number of times, you will get a ratio of exactly 1:2, no matter what. You must - it's infinity. If you do the calculations, the probability of all tosses coming up heads tends towards zero. Take it to infinity, and you get exactly zero. ie, impossible. If you take the likelihood of half heads half tails to infinity, you get one, ie, certainty.


I do poi nearly every day. But it's not like I'm addicted or anything. I mean, sure, I am always conscious of exactly where my poi are at all times, but I'm not obsessed. um.. Anyone have the number for Poi-ers Anon?

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:quote:Originally posted by SlightlySinged:
hahaha Sorry, Coleman, I didn't read your post very well, obviously i do it all the time

quote:
Okay, so if the universe is of finite size, what's one meter further away?you could consider the universe as finite if you go back to the old ideas of toroidal structure and so on.
please don't think i'm suggesting you do that though!


quote:If the universe is of infinite size, then it must be of infinite mass. your logic here is questionable in the extreme - what is the justification for this statement?
why should infinite space imply infinite energy (mass)?


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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zeuya
BRONZE Member since Oct 2003

zeuya

member
Location: Manchester

Total posts: 22
Posted:...and i just came here to learn poi! now i can get a degree in Infinity studies! ....

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simian


simian

110% MONKEY EVERY TIME ALL THE TIME JUST CANT STOP THE MONKEY
Location: London

Total posts: 3149
Posted:and witness frenetic geek ruckus. Don't dismiss frenetic geek ruckus.

I'm looking forward to gloating in Matthew BM's face at spitz tonight. Ah piddy the foo' that can't adequately explain Xeno's Paradox.

i'm currently storing up energy reserves to launch into a bitter second assault on the subject of probabilities in relation to infinity.

ooh, just you wait till i don't have anything better to do.

*shakes fist defiantly*


"Switching between different kinds of chuu chuu sometimes gives this "urgh wtf?" effect because it's giving people the phi phenomenon."

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:look at that damn ape shakin his fist at us learned scientists - its descent of the highest calibre.

i'm concerned at his progress though - he's evolving right before our eyes i tell you.
our understanding of infinities is the only thing that separates us from his a-team-rotted-double-staff-addled-monkey-brain.

well, that and the fact that i rarely pick fleas from my brother's hide and eat them (i've tried but every time i reach out, i just halve the distance between my hand and the flea and can ever seem to actually reach it...?).

i wait with glee to see mr simian will attempt to define 17.5% of infinity.
its all a ploy by the govenrment - they got a whole bunch of monkeys trying to work out vat on everything ever.
as this amount tends to infinity, they need that figure too.
and simian is the monkey to do it for them

having said that, if he thinks to pick up a twig, i'm running for the f*ckin hills...


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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SlightlySinged
GOLD Member since Jun 2003

member
Location: Melbourne, Australia

Total posts: 82
Posted:Surely if an infinite space has some energy, it must have infinite energy. Why would there be a limit? If the space never ends, NEVER ENDS, then you say that all the mass is concentrated in our own infinitely small part of the universe? I find that harder to believe. In an infinite space, if something has the chance to happen once, it must have the chance to happen over and over again, infinitely. Okay, even if you don't believe galaxies similar to ours exist right now, what about backwards and forwards in time? Is time infinite? Man, infinity really makes me feel small... It's almost enough to make me convert to religion. Almost.

Here's another thought I just had - infinite velocity means zero time, ie instant displacement, over any distance. Also, infinite acceleration is exactly the same as infinite velocity. Of course, we all believe that velocity doesn't exist past the speed of light, since time slows down and screws the whole thing up. Interesting thought, though, I think hehe. You can even take it to angular velocity, in which case if something is moving in a CIRCLE at infinite velocity, then it is at every point in the circle simultaneously hehehe. Now that's some serious poi spinning.


I do poi nearly every day. But it's not like I'm addicted or anything. I mean, sure, I am always conscious of exactly where my poi are at all times, but I'm not obsessed. um.. Anyone have the number for Poi-ers Anon?

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coleman
SILVER Member since Aug 2002

coleman

big and good and broken
Location: lunn dunn, yoo kay

Total posts: 7330
Posted:okay...

well, i can't *prove* you wrong cos i can't measure the total mass of the universe (i'd be very clever and well famous if i could).
still, your argument is very strange.

if there was infinite mass, surely all the 'space' in the universe would be filled up by it - if there is infinite mass, why all the gaps...?

there is also the small problem of the second law of thermodynamics which doesn't work for infinite energies but has been successfully used to build today's cosmological models.

i don't disagree that there are millions of galaxies that exist, and that many of these are similar to ours, but this is not an argument for infinite mass.

the 'is time infinite' question we'll leave to another thread

[ 12. November 2003, 23:53: Message edited by: coleman ]


"i see you at 'dis cafe.
i come to 'dis cafe quite a lot myself.
they do porridge."
- tim westwood

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onewheeldave
GOLD Member since Aug 2002

Carpal \'Tunnel
Location: sheffield

Total posts: 3252
Posted:A mathematician called Cantor proved some interesting stuff about infinite sets, namely that some infinite sets were 'bigger' than others.

The arguments he used are pretty accesible and easy to understand.

----------------------------------

There are two ways to count the members of a set.

Suppose you need to know how many people are in a lecture theatre; one way is to do a head count '1,2,3..' etc, till each person is accounted for and you have a final figure.

But suppose you can see that each seat is occupied and you happen to know that there are exactly 50 seats; from this you can know there are 50 people without having to do a head count.

This second way of counting is the process of seeing that the members of one set (people), can be placed in a 'one to one' correspondence with the members of another (seats).

Cantor realised that infinite sets couldn't be counted by the first method, but that the second method could be used.

To make this clear he compared the infinite set of integers (1,2,3....) with the infinite set of even numbers (2,4,6.....).

As the set of integers contains all the even numbers and all the odd numbers as well, it would be natural to think that it is the larger set.

Cantor, however, demonstrated that the two sets were equal, because they could be put in a one to one correspondence with each other, as follows: -

1 2 3 4 5 6 .....
2 4 6 8 10 12....

------------------------

Next, he asked if the set of integral fractions (1/2, 1/6, 222/4567 etc) was larger or equal to the set of whole numbers.

He showed that they could be placed in one to one correspondence and thus that the set of fractions was not larger than the set of whole numbers.

First he showed how the set of all fractions could be generated: -

1/1 1/2 1/3 1/4....

2/1 2/2 2/3 2/4....

3/1 3/2 3/3 3/4.....

4/1 4/2 4/3 4/4....

i.e. it continues to infinite extent in both horizontal and vertical directions; it contains duplicates (e.g. 1/1, 2/2/, 3/3 are all the same number) but covers all possible integral fractions.

Having generated the sequence we can now draw a wriggly line from the top corner that weaves diagonally through the entire set i.e. it would go: -

1/1 1/2 2/1 3/1 2/2 1/3 2/3 .... etc

The numbers on this line can be placed in one to one correspondence with the set of whole numbers, therefore the set of integral fractions is not larger than the set of whole numbers.

----------------------------

Cantor then asked if there was a set that was bigger than the set of whole numbers and determined that there was such a set- the set of all decimal fractions.

This includes such numbers as 0.5 1.25 3.3333... as well as the irrational numbers like Pi. The irrationals cannot be expressed as integral fractions like 3/5, but only as infinitly long decimal fractions.

To demonstrate that the set of decimals can't be put in one to one corespondence with whole numbers, Cantor used the method of assuming that they could and deriving a contradiction from that assumption, thus showing that the assumption is incorrect.

He imagined a complete list of decimal fractions placed in one to one correspondence with the integers (1,2,3....): -

1---- 0.236744.......
2---- 0.673334.........
3---- 0.999768..........
4---- 0.567321......
5---- 0.556678.........

This set would extend downwards infinitly.

For the sets to be equal, the list would have to be complete; Cantor now demonstrated that there is at least one decimal not contained in the list i.e. that it is not complete.

He did this by drawing an imaginary diagonal line passing from the first digit of the first number on the list, through the sencond digit of the second number etc etc.

Using the list above, this would generate the number: -

0.27937.......

He then constructed a number from this one by going to each digit and altering it

e.g

0.78658...

This second number would therefore differ in at least one of its digits from each of the numbers on the list that we were assuming was complete.

This demonstrates that the list was in fact incomplete, and that the set of decimal fractions is larger than the set of whole numbers.
=====================


"You can't outrun Death forever.
But you can make the Bastard work for it."

--MAJOR KORGO KORGAR,
"Last of The Lancers"
AFC 32


Educate your self in the Hazards of Fire Breathing STAY SAFE!

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