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sabianq
2010-Feb-11, 04:46 AM
it occurred to me that one can have an infinite set of numbers that is larger than another set of infinite numbers.

here is an example,
i can have the number set [1,2,3,4...]
and i can have the set [2,4,6,8...]

both sets are infinite sets
but the set on the bottom is missing every other number
so the set on the bottom is half the size.
but both sets are infinite...

Jens
2010-Feb-11, 04:53 AM
I think that's been discussed here before, but it's certainly not difficult to find explanations of that kind of thing in books and on the web. There are times when you have something that is infinity times bigger than another infinite number. For example, the integers are infinite in number, but there is an infinite number of real numbers between each integer, so the number of real numbers is infinitely greater than the already infinite number of integers.

sabianq
2010-Feb-11, 05:02 AM
meaning that the whole can be greater than the sum of its parts?

01101001
2010-Feb-11, 05:35 AM
but the set on the bottom is missing every other number
so the set on the bottom is half the size.

So... you think there's no bijection (http://en.wikipedia.org/wiki/Bijection) between the sets?

Grey
2010-Feb-11, 05:42 AM
Actually, it's trickier than you might think to compare the sizes of infinite sets. You can't just count all the items in the sets, since they are infinite. What is done is to see if you can determine a one-to-one correspondence between the two sets, matching up each element of one with an element of the other, with none left over on either side. If you can do that, the sets are equal in size; we say they have the same cardinality. The set of even numbers actually has the same cardinality as the set of all integers; they're the same size, even though it seems like one is bigger. Interestingly, both of those are also the same size as the set of rational numbers (numbers that can be represented by a finite fraction, like 5/7) and the set of algebraic numbers (numbers that are the solution to a finite polynomial equation, so this set includes things like the square root of 2).

But even though the example you give is not one of them, you are correct that not all infinities are equal. Jens is right, there are more real numbers than integers.

publius
2010-Feb-11, 05:57 AM
Infinity plays tricks on the mind. The set of integers is no bigger than the set of even or odd integers. See Hilbert's Hotel:

Now, the set of real numbers, however, is "bigger" than the set of integers. Integers are "countably infinite". The points on the real line are incountably infinite. :)

There is a progression of infinities. See the work of Cantor and his aleph numbers.

-Richard

Ivan Viehoff
2010-Feb-11, 09:41 AM
What blows my mind is the following:

You can have a denumerable subset of the reals that is dense* (eg the rationals)

And

You can have a nondenumerable subset of the reals that is nowhere dense**, eg the Cantor set

*If a subset A of the reals is dense in an interval [a,b], it means that between any two real numbers in [a,b] there is an element of A. This implies that the topological closure (under the Euclidean topology) of the part of A lying in [a,b] is precisely [a,b].

** If a subset A of the reals is nowhere dense, it means that there is no interval where A is dense.

Moose
2010-Feb-11, 12:17 PM
both sets are infinite sets
but the set on the bottom is missing every other number
so the set on the bottom is half the size.
but both sets are infinite...

1) Yes.
2) Yes.
3) No. The set on the bottom is half as dense, but they're the same size: infinite.
4) Yes.

Infinity is weird. Don't expect it to match your instinctive expectations. It rarely will.

Swift
2010-Feb-11, 06:32 PM
I think that's been discussed here before, but it's certainly not difficult to find explanations of that kind of thing in books and on the web. There are times when you have something that is infinity times bigger than another infinite number. For example, the integers are infinite in number, but there is an infinite number of real numbers between each integer, so the number of real numbers is infinitely greater than the already infinite number of integers.
Exactly. These different kinds of infinity are referred to by something called their Alelph number (http://en.wikipedia.org/wiki/Aleph_number) (Aleph zero, Aleph one).

An Alpeh Zero infinity is a "countable" infinity, such as the set of positive integers (1, 2, 3...). If you had an infinity amount of time you could count them.

An Alpeh One infinity is a non-countable infinity, such as the number of real numbers between 0 and 1. There is not even a thought experiment way to count all of them.

There are higher Alpeh number infinities, but at that point I'm lost.

LookingSkyward
2010-Feb-11, 06:37 PM
My brain is full, can I go home now?

sabianq
2010-Feb-11, 06:39 PM
is not the number 1 (one) smack in the middle of (zero) 0 and infinity?
it is the only number in the whole universe that you can multply and divide by itself and not have the answer either zoom off to zero (0) or zoom off to infinity,
depending on what side of one (1) you start on...

Nick Theodorakis
2010-Feb-11, 06:41 PM
The thing that blew my mind was when I learned that the set of rational numbers is both densely ordered (i.e., between any two rational numbers there is another) yet countable.

The wiki has a diagram showing a thought experiment to count all the rational numbers.

http://en.wikipedia.org/wiki/Rational_number#Properties

Nick

kleindoofy
2010-Feb-11, 08:45 PM
If infinity^2 were defined, I'd order a dozen.

But it's not.

So I can't.

(I think I just saved myself some cash.)

RAF_Blackace
2010-Feb-11, 08:55 PM
it occurred to me that one can have an infinite set of numbers that is larger than another set of infinite numbers.

here is an example,
i can have the number set [1,2,3,4...]
and i can have the set [2,4,6,8...]

both sets are infinite sets
but the set on the bottom is missing every other number
so the set on the bottom is half the size.
but both sets are infinite...

Yes. I watched the Horizon program on the BBC last night as well. :whistle:

Van Rijn
2010-Feb-11, 10:40 PM
This thread reminds my of the Michael McCollum story, A Greater Infinity. He's not the best writer, but there are some scenes from that story I still remember. Also, I always liked the title.

sabianq
2010-Feb-11, 11:17 PM
Yes. I watched the Horizon program on the BBC last night as well. :whistle:

actually, that is one i missed...
probably out shovelin snow...

i was counting snow flakes

Jens
2010-Feb-12, 02:47 AM
i was counting snow flakes

I assume you didn't make it to infinity. :)

Also, I don't think it's accurate to say that 1 is in the "middle" of zero and infinity, because I don't think there is such a thing as a middle in that case. Though I sort of understand what you're getting at, that 1 is a special number. I suppose that it (and zero) are sort of special. I wonder if it would be fair to say that zero is half-way between negative infinity and positive infinity?

sabianq
2010-Feb-12, 03:37 AM
if you can say that (0) zero is equal to infinity, then 1 would be in the center..

i mean, you cant really ever reach zero, you have to go way down past 1x10^-10000000000000000000000000000000000000000000000000 0000 before you ever even come close to reaching zero,
and even such a number is infinitely far from zero.

it seems to me that zero is not the same as "nothing" as "nothing" is an absence of something and zero is and unmeasurable something..

if i have a photon, its frequency cannot be zero because its wavelength would be infinite...

an absence of a photon would not be defined as a photon with a wavelength of infinity rather it would simply be an absence of the photon.

like saything something is 100%, it is a misnomer, rather it is 99.99999999, still not even close to 100%

zero divided by anything is zero.
anything divided by zero is "undefined"
however, infinity can be misused..

(I represents Infinity)
If I + I = I and I + 3 = I
Then I + I = I + 3
So I = I + 3 - I
Then I = 3

but you could also say that
Since I + 3 = I and I - 5 = I
Then I + 3 = I - 5
So 3 + 5 = 0
Then 3 = -5

(saw this creative accounting on some left over paperwork by some really big bank) [/snark]
this also shows that infinity is a variable (or can be treated an a variable)

as for the snow flake thing,
if 1 cubic foot of snow is equal to about a billion flakes, then a cubic yard of snow is about 9 billion flakes...

just take the square milage of the covered land and find out what the avarage snow fall was for the area.
find out what the total amount of snow is in the area, convert it to cubic yards and multiply that by 9 billion..

im trying to find the numbers when i have some time...

:)

sabianq
2010-Feb-12, 03:46 AM
one is the only number the universe that seems to orbit between zero and infinity.
the multiply/divide trick does show that the number 1 is indeed a very special number.

Jens
2010-Feb-12, 04:00 AM
(I represents Infinity)
If I + I = I and I + 3 = I
Then I + I = I + 3
So I = I + 3 - I
Then I = 3

I think what that mostly shows is not that infinity is a variable, but rather than you can't do arithmetic with infinity since it's not a number.

Jens
2010-Feb-12, 04:07 AM
if you can say that (0) zero is equal to infinity, then 1 would be in the center..

Whoever said that? I can't see it would be.

i mean, you cant really ever reach zero, you have to go way down past 1x10^-10000000000000000000000000000000000000000000000000 0000 before you ever even come close to reaching zero,
and even such a number is infinitely far from zero.

It sounds like you're bringing Zeno's paradox into this but I'm not sure why. We're not talking about motion, are we? If you use that logic, then you can never get to 2 either, because it's just 1.9999999...

like saything something is 100%, it is a misnomer, rather it is 99.99999999, still not even close to 100%

No, if we're talking about numbers, then 100% is 100%. 3 is 100% of 3. I don't know why you have to "get to" a number. You could come from the other side I suppose, and say it's 100.0000000000...1%. But why are you bringing movement into this? That's what I don't understand.

Jeff Root
2010-Feb-12, 06:09 AM
One is certainly a very special number, but it is no more midway between
zero and infinity than any other number.

Likewise, zero is certainly a very special number, but it is no more midway
between positive infinity and negative infinity than any other number.

-- Jeff, in Minneapolis

Jens
2010-Feb-12, 06:25 AM
Likewise, zero is certainly a very special number, but it is no more midway
between positive infinity and negative infinity than any other number.

Yeah, that makes sense. On a very naive level it seems like it should be in the middle, but obviously, if we take say 10,000, there are an infinite number of numbers to the right and an infinite number to the left. So it is in the middle in the same way.

Jeff Root
2010-Feb-12, 07:02 AM
This made me think of a little measurement conundrum that I have posted
(somewhere) previously:

I have three steel rods of different lengths: 1 cm, 40 cm, and 100 cm.
Is the 40 cm rod closer in length to the 1 cm rod or the 100 cm rod?

-- Jeff, in Minneapolis

Jens
2010-Feb-12, 07:08 AM
I have three steel rods of different lengths: 1 cm, 40 cm, and 100 cm.
Is the 40 cm rod closer in length to the 1 cm rod or the 100 cm rod?

Doesn't that stem from an ambiguity about what it means to be close? One method would be to find the average length of the two, and see which side it falls of the average. In that measurement it's closer to 1. But on the other hand, it's 40 times larger than 1, but only slightly under half of 100.

HenrikOlsen
2010-Feb-12, 12:46 PM
I have three steel rods of different lengths: 1 cm, 40 cm, and 100 cm.
Is the 40 cm rod closer in length to the 1 cm rod or the 100 cm rod?
Yes.
Which it is depends on which definition of "close" you chose to use.

01101001
2010-Feb-12, 03:16 PM
1 cm, 40 cm, 100 cm?

Plotting lengths on a linear number line, or a logarithmic?

Or something really exotic? Then you can get whatever answer you want. 40 cm is midway -- if you want it.

tashirosgt
2010-Feb-12, 05:02 PM
The "midway" discussion moves us from the territory of the transfinite cardinals to the the land of the transfinite ordinals. There are modern physical theories that involve many dimensions. Are there any that use exotic orderings?

peteshimmon
2010-Feb-12, 06:29 PM
I think people should know that one
distinguished mathematician is recorded as
going round the twist thinking about
this subject! So be careful.

On another note, it might be of interest to
consider that all of existence is an outcome
of the logical impossibility of complete zero
coupled with infinity. If you know what I mean.

Dont ask me to elaborate, thats as far as I go.

:)

cosmocrazy
2010-Feb-12, 08:32 PM
I think people should know that one
distinguished mathematician is recorded as
going round the twist thinking about
this subject! So be careful.

On another note, it might be of interest to
consider that all of existence is an outcome
of the logical impossibility of complete zero
coupled with infinity. If you know what I mean.

Dont ask me to elaborate, thats as far as I go.

:)

You don't need to, infinity and zero are both physical impossibilities in the universe we observe, If you want to try and physically define them then you could say they are one and the same thing exactly opposite to and exactly equal to each other dependent on your definition. Mathematically one can do what one likes to, with either term.

sabianq
2010-Feb-13, 12:06 AM
No, if we're talking about numbers, then 100% is 100%. 3 is 100% of 3. I don't know why you have to "get to" a number. You could come from the other side I suppose, and say it's 100.0000000000...1%. But why are you bringing movement into this? That's what I don't understand.

i am talking about the numbers in the real world.
like what was pointed out above.

You don't need to, infinity and zero are both physical impossibilities in the universe we observe,

like the speed of light..
you can be 99.999999999999999999... but can never actually reach the speed of light (100%)
you cannot ever have 100% efficiency..

likewise, you cannot ever have a photon with a wavelength of zero. if it were, the frequency would be infinite.

if you had a wavelength on infinity.
then the frequency would be zero.

the only way such a scenario is realized is when there is an absence of a photon.

sabianq
2010-Feb-13, 12:10 AM
in the real world also
if i cut a pole to 40cm,
is the length 'exactly 40cm? what is your margin of error? .0000000000000000000001cm?
the margine of error says where the length fits a tolerance..
in reality, you can never ever ever have a piece of iron that is 40.00000000000000000000000....cm long.

HenrikOlsen
2010-Feb-13, 12:53 AM
Actually, Zero is quite acceptable in this universe as there are plenty things that don't exist, the most unlikely number is probably One.

2010-Feb-13, 12:55 AM
in reality, you can never ever ever have a piece of iron that is 40.00000000000000000000000....cm long.

Well, you can, but you could never prove it. (And it won't last long due to temperature changes.)

sabianq
2010-Feb-13, 12:59 AM
Actually, Zero is quite acceptable in this universe as there are plenty things that don't exist, the most unlikely number is probably One.

lol
i have been thinking about that too..

i cant think of a single thing in nature where there is only 1 (one) of..
you can say that there are unique permentations of something like a personality, or the crystaline structure of a snowflake, or a piece of art..

but there are many snowflakes, many pieces of art etc..

if the number 1 (one) is so eloosive, then i would have to ask why people think there was only one "Big Bang"...

HenrikOlsen
2010-Feb-13, 01:13 AM
i cant think of a single thing in nature where there is only 1 (one) of..
The people we shouldn't comment too much on says there's only one planet with life in the universe.

sabianq
2010-Feb-13, 01:29 AM
The people we shouldn't comment too much on says there's only one planet with life in the universe.

yea... that just seems like a silly concept to me...
only one planet with life...

makes no sense to me..
sorry...

hewhocaves
2010-Feb-13, 01:32 AM
lol
i have been thinking about that too..

i cant think of a single thing in nature where there is only 1 (one) of..
you can say that there are unique permentations of something like a personality, or the crystaline structure of a snowflake, or a piece of art..

but there are many snowflakes, many pieces of art etc..

if the number 1 (one) is so eloosive, then i would have to ask why people think there was only one "Big Bang"...

Indeed, one is the loneliest number.

Otherworldly
2010-Feb-13, 08:05 AM
1 cm, 40 cm, 100 cm?

Plotting lengths on a linear number line, or a logarithmic?

Or something really exotic? Then you can get whatever answer you want. 40 cm is midway -- if you want it.

You could even have 100 cm half way between 1 cm and 40 cm with sufficient creativity.

Otherworldly
2010-Feb-13, 04:12 PM
You don't need to, infinity and zero are both physical impossibilities in the universe we observe,

Then I observe physical impossibilities all the time.

hhEb09'1
2010-Feb-13, 05:06 PM
i cant think of a single thing in nature where there is only 1 (one) of..Just one universe...

And I'm absolutely certain that there is just one Me. :)

Otherworldly
2010-Feb-13, 05:50 PM
lol
i have been thinking about that too..

i cant think of a single thing in nature where there is only 1 (one) of..

See below.

one is the only number the universe that seems to orbit between zero and infinity.

sabianq
2010-Feb-13, 06:06 PM
Just one universe...

And I'm absolutely certain that there is just one Me. :)

your statment falls into what i already pointed out:

i cant think of a single thing in nature where there is only 1 (one) of..
you can say that there are unique permentations of something like a personality, or the crystaline structure of a snowflake, or a piece of art..
but there are many snowflakes, many pieces of art etc..

many humans...

sabianq
2010-Feb-13, 06:09 PM
See below.

yea, but i am talking about real things..

galaxies, stars, rocks, spiders, worms, zebras, atoms, quarks, bosons, leptons, photons...

hhEb09'1
2010-Feb-13, 06:11 PM
your statment falls into what i already pointed out:
Many universes? :)

Even multiverses can still be classified as a single universe.

sabianq
2010-Feb-13, 06:12 PM
and just because there is lack of proof that there is only one universe, that lack of proof does not in anyway prove that there is only one universe..

or am i wrong on this?

sabianq
2010-Feb-13, 06:13 PM
semantics..

m-theory suggests that there are indeed many universes all with their own bubbles of time/space

sabianq
2010-Feb-13, 06:14 PM
http://en.wikipedia.org/wiki/Multiverse

hhEb09'1
2010-Feb-13, 06:14 PM
yea, but i am talking about real things..

galaxies, stars, rocks, spiders, worms, zebras, atoms, quarks, bosons, leptons, photons...real classifications!

What about BAUT users named sabianq? There's only one of those. Sure there are many users.

But there are many things, and all things are things. There's only one category that contains all things. :)

sabianq
2010-Feb-13, 06:17 PM
real classifications!

What about BAUT users named sabianq? There's only one of those. Sure there are many users.

But there are many things, and all things are things. There's only one category that contains all things. :)

Originally Posted by sabianq
i cant think of a single thing in nature where there is only 1 (one) of..
you can say that there are unique permentations of something like a personality, or the crystaline structure of a snowflake, or a piece of art..
but there are many snowflakes, many pieces of art etc..

many members
many names
many catagories
many forums

peteshimmon
2010-Feb-13, 06:19 PM
Absence of evidence is not evidence of absence.

click...sound of running..

hhEb09'1
2010-Feb-13, 06:22 PM
and just because there is lack of proof that there is only one universe, that lack of proof does not in anyway prove that there is only one universe..

or am i wrong on this?Yes, if you use the right definition of universe. :)

sabianq
2010-Feb-13, 06:24 PM
like i said...

semantics..

hhEb09'1
2010-Feb-13, 06:42 PM
In which one post did you say that? :)

Jeff Root
2010-Feb-13, 07:21 PM
Sabianq,

Getting agreement on the meanings of terms is essential. Redefining terms
arbitrarily to make your statements true is what you mean by 'semantics'.
(Which, perhaps, I did right there. :-)) You are the person redefining terms
in order to make true your comment about non-uniqueness. My opinion is
that the comment has some truth it, but not much. Its truth is dependant
on how you define your terms. So it says very little about reality and more

-- Jeff, in Minneapolis

sabianq
2010-Feb-13, 10:45 PM
Redefining terms arbitrarily to make your statements true is what you mean by 'semantics'.

Universe...
the visible univese.
what we can see with out telescopes..
what was created by the big bang..

i didnt change the mainstream meaning Jeff..

sabianq
2010-Feb-13, 10:48 PM
post 47

semantics..

m-theory suggests that there are indeed many universes all with their own bubbles of time/space

sabianq
2010-Feb-13, 10:50 PM
it was hhEb09'1 that changes the meaning of Universe from the classical mainstream meaning of a universe that was created from a big bang..

Many universes? :)

Even multiverses can still be classified as a single universe.

that was post 45
in post 47,
i said

Originally Posted by sabianq
semantics..

sabianq
2010-Feb-13, 10:57 PM
Sabianq,

You are the person redefining terms in order to make true your comment about non-uniqueness. -- Jeff, in Minneapolis

no true at all,
it was the other poster that changed the meanig of multiple universes to a single all encompasong universe to underscore non-non-uniqueness..

this is silly,
my infinity is still bigger than yours..
:lol:

tommac
2010-Feb-14, 03:11 AM
Infinity plays tricks on the mind. The set of integers is no bigger than the set of even or odd integers. See Hilbert's Hotel:

Now, the set of real numbers, however, is "bigger" than the set of integers. Integers are "countably infinite". The points on the real line are incountably infinite. :)

There is a progression of infinities. See the work of Cantor and his aleph numbers.

-Richard

Isnt the set of lines bigger than the set of real numbers and the set of curves bigger than the set of lines? I think i remember reading bout that once ...

hhEb09'1
2010-Feb-14, 04:59 AM
no true at all,
it was the other poster that changed the meanig of multiple universes to a single all encompasong universe to underscore non-non-uniqueness..sooooo...you're saying, that each multiverse is not unique? In other words, they're all the same? Doesn't that mean, there's only really one?

this is silly,yeppers!

sabianq
2010-Feb-14, 05:25 AM
well, you can say that there are many multiverses!
and they all reside on one 11 dimensional universe, which may be but a mere 11 dimensional single bubble of a whole universe in a much vaster multiverse which is itself a bubble of an even bigger universe which is only a itty bitty tiny bubble in a foam of undulating dimensional chaos all within an overall universe that itself is a bubble in...

lol,

i guess that is the definition of infinity and forever..

and silly can be fun and amusing and thought provoking...

cosmocrazy
2010-Feb-14, 11:07 AM
Then I observe physical impossibilities all the time.

Nope, you can't "observe" zero anything likewise you can't "observe" infinite anything either...good luck! :D

cosmocrazy
2010-Feb-14, 11:14 AM
Actually, Zero is quite acceptable in this universe as there are plenty things that don't exist, the most unlikely number is probably One.

Depends on what you are defining and what you believe in. 1 me, 1 you, 1 universe, 1 creator....and so on. Yes there could be many variations of the same things and it appears that lots of things come in pairs i.e matter/antimatter and such like. But there are countless unique things we observe everyday.

"there can be only one....!" http://www.youtube.com/watch?v=kq4SqgxIKM0 (Highlander). compliments of chrissy. :)

Otherworldly
2010-Feb-14, 11:39 AM
Nope, you can't "observe" zero anything likewise you can't "observe" infinite anything either...good luck! :D

I have no problem observing the absence of something. Good luck proving otherwise!

HenrikOlsen
2010-Feb-14, 01:00 PM
Isnt the set of lines bigger than the set of real numbers and the set of curves bigger than the set of lines? I think i remember reading bout that once ...
The sets of cartesian products of real numbers has the same cardinality as the set of real numbers, and as a line (in a plane) can be defined uniquely by two numbers, the cardinality of the set of all lines is the same as that of the real numbers (Aleph_1).
The cardinality of the set of all curves on the other hand is Aleph_2 as far as I remember.

cosmocrazy
2010-Feb-14, 02:29 PM
I have no problem observing the absence of something. Good luck proving otherwise!

Ah... I see you point now, we seem to be interpreting things differently. Yes observing the absence of a specific object of your choice one can do. :)

tommac
2010-Feb-14, 04:33 PM
The sets of cartesian products of real numbers has the same cardinality as the set of real numbers, and as a line (in a plane) can be defined uniquely by two numbers, the cardinality of the set of all lines is the same as that of the real numbers (Aleph_1).
The cardinality of the set of all curves on the other hand is Aleph_2 as far as I remember.

I read this kind of cool book "1,2,3 infinity" George Gamow a long while back that was all about this stuff. Of course I forgot it all. I remember it being a great book and very interesting read.

One Two Three . . . Infinity: Facts and Speculations of Science (Paperback) (http://www.amazon.com/exec/obidos/ASIN/0486256642/atkinsdietmailin)

mike alexander
2010-Feb-14, 05:23 PM
The section on (I think) Cantor in JT Bell's Men of Mathematics has a nice explanation of the diagonal proof, showing how there can be different levels of infinity.

publius
2010-Feb-14, 08:59 PM
Isnt the set of lines bigger than the set of real numbers and the set of curves bigger than the set of lines? I think i remember reading bout that once ...

Sorry, but I just don't know. Aleph_2 would of course be the next one, and I have no idea what sets have that cardinality.

-Richard