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Code Red
2002-Jun-10, 12:28 PM
An alcohol-induced discussion between a friend and myself last night -

Friend - The concept of Infinity encompasses all that has ever existed and ever will exist.

Me - You have a cylinder, diameter x and length infinity. Therefore it has infinite volume. Next to it is another cylinder, diameter 2x and length infinity. Therefore it also has infinite volume. The second cylinder is certainly the bigger of the two, but they are both infinite in volume, are totally separate bodies and both exist within a larger space which is also infinite. Therefore the concept of infinity as a mathematical value is relative.

Friend - Pass the Coors.

Which of us is talking the bigger load of crap?

_________________
Red

"Go to Blue Alert!"
"Is that really necessary Sir? It will mean changing the bulb..."

<font size=-1>[ This Message was edited by: Code Red on 2002-06-10 08:30 ]</font>

Wally
2002-Jun-10, 12:51 PM
Well. . . you!

Your friend's last statement pretty much summarizes all of life's mysteries quite nicely, I think. /phpBB/images/smiles/icon_smile.gif

2002-Jun-10, 01:04 PM
Hey, Red!

Maybe you should have invited Kantor to the discussion.

Code Red
2002-Jun-10, 01:25 PM
Your friend's last statement pretty much summarizes all of life's mysteries quite nicely, I think. /phpBB/images/smiles/icon_smile.gif

Funny, I was thinking pretty much the same thing...

David Hall
2002-Jun-10, 02:35 PM
I think the subject of infinite sets has been pretty well thought out by mathematicians. In spite of a set being "infinite", there's always a way to make it larger. an infinite set can always be added to x+1. In addition, a set consisting only of the even numbers is larger than a set consisting of whole integers...or is it? A set of only prime numbers is even bigger, as is a set of all irrational numbers. And then what about x*2?

Does an infinite 3 dimensional set have more points than an infinite 2 dimensional set? See, we can continue like this infinitely. /phpBB/images/smiles/icon_biggrin.gif

_________________
David Hall
"Dave... my mind is going... I can feel it... I can feel it." (http://www.occn.zaq.ne.jp/cuaea503/whatnots/2001_feel_it.wav)

<font size=-1>[ This Message was edited by: David Hall on 2002-06-10 10:39 ]</font>

David Hall
2002-Jun-10, 02:37 PM
Oh, and so, in conclusion, I also agree with your friend. It's too much for one little mind to grasp. Might as well grasp a beer instead. /phpBB/images/smiles/icon_smile.gif /phpBB/images/smiles/icon_biggrin.gif

SeanF
2002-Jun-10, 02:44 PM
On 2002-06-10 10:35, David Hall wrote:
In addition, a set consisting only of the even numbers is larger than a set consisting of whole integers...or is it?

Huh? I can see how one might argue that the whole integers set is larger than the even numbers set, but how do you argue that it's smaller?

(BTW, I've always argued that since there's an exact 1-1 ratio of integers to even integers [1=2, 2=4, 3=6, 4=8, 5=10 . . . ] the two infinite sets must, in fact, be equal in size.) /phpBB/images/smiles/icon_smile.gif

Wiley
2002-Jun-10, 02:59 PM
Maps!

It all comes down to the maps. Instead of a cylinder consider an infinitely long thin strip. Why a strip as opposed to a cylinder? The equations are simpler to write in a text only setting.

We have two infinitely long strips; the width of the first is from x = 0 to x = 1 while the width of the second is from x = 2 to x = 4. The function f(x) = 2*x + 2 will take a point on the first strip and find a corresponding point on the second strip. This is a one-to-one mapping; in other words, for every point on the first strip there is a unique point on the second strip. An important feature of a one-to-one map is that we can invert it. The function g(x) = f^(-1)(x) = x/2 - 1 will take a point on the second strip and find the corresponding point on the first strip.

If a one-to-one map exists between two infinities, these infinities are said to be "equal". There are two known types of infinities, countable (denoted by aleph sub 0) and uncountable (denoted by aleph sub 1). Countable infinite sets map to the set of integers while uncountable map to the set of irrationals. If you find another type of infinity, you're well on your way to proving the "continum hypothesis" and earning a million dollars.

P.S. Put down the Coors, there's better Colorado beer: Fat Tire, Left Hand ....

<font size=-1>[ This Message was edited by: Wiley on 2002-06-10 11:13 ]</font>

David Hall
2002-Jun-10, 03:03 PM
On 2002-06-10 10:44, SeanF wrote:

Huh? I can see how one might argue that the whole integers set is larger than the even numbers set, but how do you argue that it's smaller?

Well, if you always add x+2, isn't that larger than adding x+1? But then again, with twice as many integers in the whole integers set, I guess it's larger after all. Or maybe the same size, as you suggested.

Uggh. See what I mean? It gives me a headache. I'm no mathematician. I'd rather count beers.

Silas
2002-Jun-10, 03:12 PM
Take the ordinary x-y plane, and graph the function y = 1/x, over x > 0.1

Now revolve the curve around the x axis. The surface of revolution is a kind of funnel-shape.

This funnel has a finite volume...but an infinite surface. You can fill it...but you can't paint it...

When my maths prof first showed me this, I asked, "What if you fill it with paint?" He grinned cheerfully and said, "Nope. It still doesn't work. That's the difference between 'infinity' and the real world."

Silas

David Hall
2002-Jun-10, 03:24 PM
Somebody once posted a great short story about the infinite hotel. It'd be nice to get that link again if anyone has it.

I have a book on my shelf called Infinity and the Mind: The Science and Philosophy of the Infinite (1982) by Rudy Rucker. It seems to try to cover all these concepts in pretty much layman's terms. But I've never been able to get through more than a few chapters. It is a bit of a dry read. But I may just give it another chance now, if I can fit it into my reading schedule.

GrapesOfWrath
2002-Jun-10, 03:52 PM
On 2002-06-10 11:12, Silas wrote:
When my maths prof first showed me this, I asked, "What if you fill it with paint?" He grinned cheerfully and said, "Nope. It still doesn't work. That's the difference between 'infinity' and the real world."

I always liked that one too!

The point at the bottom of the "bottle" where it is too narrow for a molecule of paint to fit below still has an infinite surface area below it.

On 2002-06-10 10:35, David Hall wrote:
I think the subject of infinite sets has been pretty well thought out by mathematicians. In spite of a set being "infinite", there's always a way to make it larger. an infinite set can always be added to x+1. In addition, a set consisting only of the even numbers is larger than a set consisting of whole integers...or is it?

No they're equal

A set of only prime numbers is even bigger,

Nope.

as is a set of all irrational numbers. And then what about x*2?

Well, the number of irrationals is bigger than the number of rationals, but I'm not sure what you mean by "x*2". Is that x times 2? What's x?

Does an infinite 3 dimensional set have more points than an infinite 2 dimensional set?

No.

See, we can continue like this infinitely. /phpBB/images/smiles/icon_biggrin.gif

Only if you come up with harder questions. /phpBB/images/smiles/icon_smile.gif

CJSF
2002-Jun-10, 03:58 PM
OW! MY BRAIN! OW OW!!!!!!!!!!

/phpBB/images/smiles/icon_smile.gif

CJSF

Code Red
2002-Jun-10, 04:08 PM
OW! MY BRAIN! OW OW!!!!!!!!!!

Excellent rubs hands - my work is done!!

nebularain
2002-Jun-10, 04:12 PM
When I took calculus, we learned to solve problems by cancelling out "infinity." That is, if the graph shows a curve that infinitely approaches a certain number, we solve the problem as if it were that number. In comparing two number as "infinity times this" or "infinity plus that," we consider both numbers to simply being "infinity" and treat them as being equal, thus cancelling out any concept of greater than or less than.

(OK, so would you consider this answer a cop-out or a kill-joy! /phpBB/images/smiles/icon_biggrin.gif )

Bob
2002-Jun-10, 04:14 PM
Speaking of beer:

Aleph one bottles of beer on the wall!
Aleph one bottles of beer!
Take Aleph null down
And pass them around!
Aleph one bottles of beer on the wall!

Wiley
2002-Jun-10, 04:58 PM
On 2002-06-10 12:14, Bob wrote:
Speaking of beer:

Aleph one bottles of beer on the wall!
Aleph one bottles of beer!
Take Aleph null down
And pass them around!
Aleph one bottles of beer on the wall!

I wish Aleph the board before I read this post. /phpBB/images/smiles/icon_smile.gif

<font size=-1>[ This Message was edited by: Wiley on 2002-06-10 13:06 ]</font>

Wiley
2002-Jun-10, 05:05 PM
On 2002-06-10 12:12, nebularain wrote:
When I took calculus, we learned to solve problems by cancelling out "infinity." That is, if the graph shows a curve that infinitely approaches a certain number, we solve the problem as if it were that number. In comparing two number as "infinity times this" or "infinity plus that," we consider both numbers to simply being "infinity" and treat them as being equal, thus cancelling out any concept of greater than or less than.

(OK, so would you consider this answer a cop-out or a kill-joy! /phpBB/images/smiles/icon_biggrin.gif )

This is l'Hopital's rule (http://www.ies.co.jp/math/java/calc/lopi/rule.html).

Kaptain K
2002-Jun-10, 05:18 PM
I wish Aleph the board before I read this post.
OUCH!!! baaaad pun. Wish I'd thought of it. /phpBB/images/smiles/icon_evil.gif

_________________
When all is said and done - sit down and shut up!

<font size=-1>[ This Message was edited by: Kaptain K on 2002-06-10 13:23 ]</font>

nebularain
2002-Jun-10, 06:13 PM
Getting back to seriousness, I found a good (at least I thought it was) expounding on infinity and aleph and such:

2002-Jun-10, 06:18 PM
On 2002-06-10 11:24, David Hall wrote:
Somebody once posted a great short story about the infinite hotel. It'd be nice to get that link again if anyone has it.

is it this (http://www.jimloy.com/calc/hotel.htm) one?

beskeptical
2002-Jun-10, 09:16 PM
I'm not going to believe in infinity, because I can't get to the end to prove it. /phpBB/images/smiles/icon_biggrin.gif

<font size=-1>[ This Message was edited by: beskeptical on 2002-06-10 17:17 ]</font>

Espritch
2002-Jun-11, 02:48 AM
I'm not going to believe in infinity, because I can't get to the end to prove it.

But if you believe in infinity, you can use it to prove all sorts of things. For instance, infinity + 1 is infinity. So infinity + infinity equals infinity. Therefore 2*infinity = 1*infinity. Devide both sides by infinity and we get 2 = 1.

Now pass me a beer.

informant
2002-Jun-11, 09:54 AM
Wiley wrote: "If you find another type of infinity, you're well on your way to proving the "continum hypothesis" and earning a million dollars."

It's easy to define an infinite number larger than the cardinality of the irrational numbers (=cardinality of real numbers).
Just think of the set of all subsets of the real line.
The set of all subsets of a particular set always has a larger cardinality than the original set, therefore you have just found a set with cardinality greater than the real line.
The issue with the continuum hypothesis is whether there is a set whose cardinality is strictly between that of the integers and that of the real numbers.

The problem with infinity is that there are so many kinds of it... If you have a problem with thinking that two sets have the same "size" just because you can find a one-to-one mapping between the two (as I sometimes do), just think of infinite cardinalities as a way to classify infinite sets.
The notion of an infinite set seems simple enough. But what Cantor and others found out is that there are different kinds of infinite sets. For instance, the integers and the rational numbers can be put into one category, whereas the irrational numbers are put into a different category.
There is also an "order" between these categories of sets, and it's in that sense that one says that there are "more" irrational numbers than there are rational numbers.

But, to address the original point,

Friend - The concept of Infinity encompasses all that has ever existed and ever will exist.

Me - You have a cylinder, diameter x and length infinity. Therefore it has infinite volume. Next to it is another cylinder, diameter 2x and length infinity. Therefore it also has infinite volume. The second cylinder is certainly the bigger of the two, but they are both infinite in volume, are totally separate bodies and both exist within a larger space which is also infinite. Therefore the concept of infinity as a mathematical value is relative.

Friend - Pass the Coors.

Which of us is talking the bigger load of crap?

I think what you and your friend did was rediscover Bertrand's paradox: that there cannot be a set which contains all sets.

<font size=-1>[ This Message was edited by: informant on 2002-06-11 05:58 ]</font>

David Hall
2002-Jun-11, 01:09 PM
On 2002-06-10 14:18, amstrad wrote:

On 2002-06-10 11:24, David Hall wrote:
Somebody once posted a great short story about the infinite hotel. It'd be nice to get that link again if anyone has it.

is it this (http://www.jimloy.com/calc/hotel.htm) one?

Well, that's a very abridged version of the full story I've seen. The full version had a lot of exteraneous detail that made it more fun to read, but the gist of the thing is here. Thanks for finding it.

And if anyone knows of a full version, please send it along.

John Kierein
2002-Jun-11, 04:40 PM
A great book that covers this is by George Gamow written back in the 50s. "1,2,3 ... Infinity" He shows that there are different classes of infinities, some of which are larger than others. There is the "number of numbers". In this you can show that the number of integers equals the number of even numbers, etc. by mapping them 1 to 1. But the "number of points on a line" is greater than the number of numbers because there are always points between any numbered points that you map a number to on a line. The infinite number of points on a line equals the infinite number of points on a surface and in a volume. The third and largest infinity is the "number of functions" because there are an infinite number of functions that can be drawn through any point on a line. Gamow's books are lots of fun, like "Mr. Tompkins in Wonderland" where he imagines what it would be like if the speed of light were only 20 mph. The Physics Bldg at the U. of Colorado is called "Gamow Tower". Gamow got Hans Bethe to co-author a paper he wrote with Alpher so it could be said it was by "Alpher, Bethe, Gamow".

Wiley
2002-Jun-11, 05:26 PM
On 2002-06-11 05:54, informant wrote:
It's easy to define an infinite number larger than the cardinality of the irrational numbers (=cardinality of real numbers).
Just think of the set of all subsets of the real line.
The set of all subsets of a particular set always has a larger cardinality than the original set, therefore you have just found a set with cardinality greater than the real line.
The issue with the continuum hypothesis is whether there is a set whose cardinality is strictly between that of the integers and that of the real numbers.

Yep, you 'ight, Wheezy. I forgot about Cantor's Theorem, which is the more formal name of what you wrote: the power set of set A has a large cardinality than A. I like to think of the continum hypothesis whether the set of cardinals is countable or uncountable. I believe this is equivalent to the definition you gave - I'll let the mathematicians figure it out.

Thanks for keeping me on my toes.

Silas
2002-Jun-11, 05:29 PM
On 2002-06-11 12:40, John Kierein wrote:
A great book that covers this is by George Gamow written back in the 50s. "1,2,3 ... Infinity" . . . Gamow got Hans Bethe to co-author a paper he wrote with Alpher so it could be said it was by "Alpher, Bethe, Gamow".

Yes indeed! A very good book! Fun, and readable.

(My uncle, Dr. Abbott, a marine biologist, always wanted to co-write a book with an oceanographer friend of his at Woods Hole named Dr. Costello. Alas, it never quite worked out!)

Silas

Chuck
2002-Jun-11, 10:42 PM
If the power set of a set is larger than the set itself, does the power set of the set of all sets contain more sets than the set of all sets?

Wiley
2002-Jun-11, 11:05 PM
On 2002-06-11 18:42, Chuck wrote:
If the power set of a set is larger than the set itself, does the power set of the set of all sets contain more sets than the set of all sets?

Now your getting into Russell's Paradox that informant mentioned. The set of all sets does not exist; it leads to contradictions. Russell and Whitehead create "type theory", or hierarchy of sets to get around Russell's Paradox.

GrapesOfWrath
2002-Jun-12, 09:57 PM
On 2002-06-11 19:05, Wiley wrote:
Now your getting into Russell's Paradox that informant mentioned.

I think you're right, but the paradox that informant mentioned was Bertand's paradox, which is something completely different. I'm sure he meant Bertand Russell's paradox.

Wiley
2002-Jun-12, 11:04 PM
On 2002-06-12 17:57, GrapesOfWrath wrote:

On 2002-06-11 19:05, Wiley wrote:
Now your getting into Russell's Paradox that informant mentioned.

I think you're right, but the paradox that informant mentioned was Bertand's paradox, which is something completely different. I'm sure he meant Bertand Russell's paradox.

Thanks, Herr Wrath. I was unfamiliar with Bertrand's Paradox, so I assumed he meant Russell's Paradox. Now that I've educated myself, I do believe he meant Russell's Paradox. I may be slow, but I fail to see the relevance of Bertrand's paradox with the concepts of infinity. Unless we have equally improbable infinities, which could occur using an Infinite Improbablity Drive.

ZaphodBeeblebrox
2002-Jun-12, 11:16 PM
On 2002-06-10 11:24, David Hall wrote:
Somebody once posted a great short story about the infinite hotel. It'd be nice to get that link again if anyone has it.

You mean this (http://www.c3.lanl.gov/mega-math/) one, don'tcha?

The Story is a Hoot, too!

ZaphodBeeblebrox
2002-Jun-12, 11:28 PM
On 2002-06-11 12:40, John Kierein wrote:
Gamow got Hans Bethe to co-author a paper he wrote with Alpher so it could be said it was by "Alpher, Bethe, Gamow".

Actually, the Joke was on Bethe, seems that Ol' Georgie, never asked for his permission, to sign his Name!

Furthermore, the Real Humour, in it, was that the Paper, was about the Beginning, of the Universe, ergo . . .

ZaphodBeeblebrox
2002-Jun-12, 11:36 PM
Unless we have equally improbable infinities, which could occur using an Infinite Improbablity Drive.

You called?

Actually, I don't really know how the darn Thing, Works, I just stole it, because it was so Amazingly Amazing!

Silas
2002-Jun-13, 02:36 AM
On 2002-06-12 19:28, ZaphodBeeblebrox wrote:

On 2002-06-11 12:40, John Kierein wrote:
Gamow got Hans Bethe to co-author a paper he wrote with Alpher so it could be said it was by "Alpher, Bethe, Gamow".

Actually, the Joke was on Bethe, seems that Ol' Georgie, never asked for his permission, to sign his Name!

Furthermore, the Real Humour, in it, was that the Paper, was about the Beginning, of the Universe, ergo . . .

Are you sure? The way I heard the story, he approached Bethe, who thought it was cute, and gave permission.

Silas

Ring
2002-Jun-13, 04:01 AM
Does this stuff have any practical significance or is it just interesting but useless pure math? Please don't get me wrong I find this stuff fascinating, but my pea brain isn't coming up with any possible uses for it.

Ring
2002-Jun-13, 04:06 AM
"useless pure math"

Sorry, that sounds pretty crappy.

John Kierein
2002-Jun-13, 10:42 AM
The infinity room is not in the hotel.
http://www.dailycamera.com/bdc/science/article/0,1713,BDC_2432_1205832,00.html

John Kierein
2002-Jun-13, 10:48 AM
On 2002-06-13 00:01, Ring wrote:
Does this stuff have any practical significance or is it just interesting but useless pure math? Please don't get me wrong I find this stuff fascinating, but my pea brain isn't coming up with any possible uses for it.

If the universe is infinite, then you need to understand some of this stuff to solve things like Olbers paradox. It turns out that energy from an infinite universe, with a red shift included, results in a finite amount of long wavelength radiation being received. It's like an infinite series that converges to a finite number, like 1/2 + 1/4 + 1/8 + 1/16 ... = < 1.

David Hall
2002-Jun-13, 04:50 PM
On 2002-06-12 19:16, ZaphodBeeblebrox wrote:

You mean this (http://www.c3.lanl.gov/mega-math/) one, don'tcha?

The Story is a Hoot, too!

Ah, thank you Big Z. that's a good link. But believe it or not, the story there seems to be quite a bit different than the one I originally encountered. It seems to me now that there are several different versions floating around the net. /phpBB/images/smiles/icon_eek.gif

Well, whatever. This one is just as good and has all of the same ideas in it, so I'll keep this one for now. Thanks again. /phpBB/images/smiles/icon_smile.gif

ZaphodBeeblebrox
2002-Jun-13, 04:54 PM
On 2002-06-13 12:50, David Hall wrote:
Well, whatever. This one is just as good and has all of the same ideas in it, so I'll keep this one for now. Thanks again. /phpBB/images/smiles/icon_smile.gif

No Problem, and if I find a Link to the Other Versions, then I'll Post Them, too!

BTW, I'm still Working, on that Bethe Quote.

Silas
2002-Jun-13, 05:27 PM
On 2002-06-13 00:06, Ring wrote:
"useless pure math"

Sorry, that sounds pretty crappy.

Grin! "Here's to modern mathematics...may it never do anybody any good!"

It may seem weird, but abstract math, which would appear to have no use at all, has led to some very practical real-world applications. Without "infinity," calculus would have no meaning...and without calculus, we wouldn't have the understanding of mechanics that took us to the moon.

"Imaginary Numbers" -- what could conceivably be more abstract? -- are used in understanding transmission of AC electricity.

Questions about infinity underlie many of our modern microchip cosmos, such as telephone switching algorithms, approximation theory, catastrophe theory (used in stock market simulations!) and so on.

So, yeh, it's useless pure math...except that it has led to a lot of actual uses!

Silas

ZaphodBeeblebrox
2002-Jun-13, 05:38 PM
On 2002-06-13 13:27, Silas wrote:
"Imaginary Numbers" -- what could conceivably be more abstract? -- are used in understanding transmission of AC electricity.

And building Truncated Pyramids, or, at least They would have been Necessary, except that The Ancient Egyptians, Fudged the Math!

Who says They weren't Like us?

Wiley
2002-Jun-13, 05:58 PM
On 2002-06-13 13:27, Silas wrote
"Imaginary Numbers" -- what could conceivably be more abstract? -- are used in understanding transmission of AC electricity.

This is an incredible understatement of the importance of imaginary numbers. It's safe to say the technological revolution would not have happened if we did not know about imaginary numbers. Imaginary numbers were a quaint oddity until Euler discovered the formula: exp(i*x) = cos(x) + i*sin(x). This led to the Fourier transform (and the Bessel, Laplace, Gaussian, Z, and other transforms).

Is there a branch of science or engineering that does not rely heavily on the Fourier transform? (No, psychology doesn't count. /phpBB/images/smiles/icon_smile.gif)

<font size=-1>[ This Message was edited by: Wiley on 2002-06-13 14:08 ]</font>

informant
2002-Jun-13, 09:28 PM
GrapesofWrath wrote:
I think you're right, but the paradox that informant mentioned was Bertand's paradox, which is something completely different. I'm sure he meant Bertand Russell's paradox.

You are correct, I said Bertrand's paradox because that's Bertrand Russell's full name.
However, I should have said Russell's paradox, since "Bertrand's paradox" (paradoxes, actually) is a paradox in probability theory. (This Bertrand is not Bertrand Russell.)