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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?

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"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>

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Well. . . you!

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

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Hey, Red!

Maybe you should have invited Kantor to the discussion.

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Your friend's last statement pretty much summarizes all of life's mysteries quite nicely, I think. [img]/phpBB/images/smiles/icon_smile.gif[/img]
Funny, I was thinking pretty much the same thing...

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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. [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

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

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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. [img]/phpBB/images/smiles/icon_smile.gif[/img] [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

7. 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.) [img]/phpBB/images/smiles/icon_smile.gif[/img]

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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>

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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.

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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

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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.

12. 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. [img]/phpBB/images/smiles/icon_biggrin.gif[/img]
Only if you come up with harder questions. [img]/phpBB/images/smiles/icon_smile.gif[/img]

13. OW! MY BRAIN! OW OW!!!!!!!!!!

[img]/phpBB/images/smiles/icon_smile.gif[/img]

CJSF

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OW! MY BRAIN! OW OW!!!!!!!!!!
Excellent rubs hands - my work is done!!

15. 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! [img]/phpBB/images/smiles/icon_biggrin.gif[/img] )

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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!

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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. [img]/phpBB/images/smiles/icon_smile.gif[/img]

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

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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! [img]/phpBB/images/smiles/icon_biggrin.gif[/img] )
This is l'Hopital's rule.

19. I wish Aleph the board before I read this post.
OUCH!!! baaaad pun. Wish I'd thought of it. [img]/phpBB/images/smiles/icon_evil.gif[/img]

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

20. Getting back to seriousness, I found a good (at least I thought it was) expounding on infinity and aleph and such:

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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 one?

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I'm not going to believe in infinity, because I can't get to the end to prove it. [img]/phpBB/images/smiles/icon_biggrin.gif[/img]

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

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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.

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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>

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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 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.

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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".

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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.

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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

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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?

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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.

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