to infinity, and then some...

[Ambrosia]

Just watched this Horizon program on infinity.

Now my head hurts.

Take the numbers 1,2,3,4,5,6,7,8,9,10 ...

Now take only the even numbers 2,4,6,8,10 ...

despite appearances, both these series of numbers are the same length

infinity * infinity = infinity
infinity + infinity = infinity
infinity / infinity = infinity
infinty - infinity = er anything you like

should you travel in a straight line (2^10¹¹⁸)*10²⁶m you will find a perfect copy of the earth and everyone and everything on it.

How far is that in light years??

Oh and the monkeys, well the chances of a monkey typing the complete works of shakespeare at random is roughly equivalent to winning the UK lottery (14millionish to 1) every week, week in, week out for 29000 years.

 

[Dilb]

Just watched this Horizon program on infinity.

Now my head hurts.

Take the numbers 1,2,3,4,5,6,7,8,9,10 ...

Now take only the even numbers 2,4,6,8,10 ...

despite appearances, both these series of numbers are the same length

Take each number in the first list, and multiply every number in it by 2. Tada, they obviously have equal elements.

Let's make it annoying: take 1,2,3,4,5,6...
Split it into two lists
1,3,5,7,...
2,4,6,8,...
Add 1 to the first list
2,4,6,8...
2,4,6,8...
And divide both by 2
1,2,3,4,5...
1,2,3,4,5...

Tada, first list is equivalent to twice itself.

What gets annoying is when you take something finite, like a ball, and divide it up into infinitely many pieces. Then you spin the pieces, put the pieces back together, and you have two balls the same size as the original.

 

[shadron]

Just watched this Horizon program on infinity.

Now my head hurts.

Take the numbers 1,2,3,4,5,6,7,8,9,10 ...

Now take only the even numbers 2,4,6,8,10 ...

despite appearances, both these series of numbers are the same length

infinity * infinity = infinity
infinity + infinity = infinity
infinity / infinity = infinity
infinty - infinity = er anything you like

should you travel in a straight line (2^10¹¹⁸)*10²⁶m you will find a perfect copy of the earth and everyone and everything on it.

How far is that in light years??

Oh and the monkeys, well the chances of a monkey typing the complete works of shakespeare at random is roughly equivalent to winning the UK lottery (14millionish to 1) every week, week in, week out for 29000 years.

Better watch it. Thoughts like this drove Georg Cantor mad.

 

[Walrus32]

Contrary to the wide-spread belief, all those monkeys would never type all the works of Shakespeare.

They would, however, type all the speeches of Barack Obama.

 

[Brian-M]

Just watched this Horizon program on infinity.

Now my head hurts.

Take the numbers 1,2,3,4,5,6,7,8,9,10 ...

Now take only the even numbers 2,4,6,8,10 ...

despite appearances, both these series of numbers are the same length

Unless I'm also confused, while both sets of numbers might be infinite, the first infinite is twice as big as the second infinite. (Not all values of infinite are the same.)

should you travel in a straight line (2^10¹¹⁸)*10²⁶m you will find a perfect copy of the earth and everyone and everything on it.

A straight line in which direction? Or are they saying we're surrounded by sphere of countless trillions of duplicate earths located exactly (2^10¹¹⁸)*10²⁶m away from us?

(There probably aren't even enough earth-like planets in the universe that you wouldn't have to be insane to expect to find one with flora and fauna that exactly match that of earth, never mind the absurdity of finding a perfect replica.)

This sounds like complete bullpoop to me, unless they're basing the claim on an (unproven) theory that the universe is curved, and you'll end up back where you started.

Oh and the monkeys, well the chances of a monkey typing the complete works of shakespeare at random is roughly equivalent to winning the UK lottery (14millionish to 1) every week, week in, week out for 29000 years.

Of course, there's no way to actually train a monkey to type in a truly random manner. Hell, you can't even train a human to type in a truly random manner. And good luck trying to force the monkey to sit there and keep typing after it get bored with typing.


Of course, if you program a computer attached to a random-number generator, and had it print-out one character at a time at approximately the same rate a monkey would be expected to type, and let it go on printing random characters for eternity, then it's almost certain that eventually come up with the complete works of Shakespeare, even if it takes a few billion times longer than the expected life of the universe to do so.

 

[Soapy Sam]

What I find strange about the concept of infinite numbers is that it is obviously inconsistent with reality, (there are no concrete examples of infinite quantities and logically cannot be, because if there exists one infinite quantity of anything then how can there be anything else?) yet at the same time they are perfectly consistent with the way we think numbers work. This seems tolerable, only because numbers are somehow "imaginary constructs" with no actual physical properties outside human minds.

To me, this suggests either we don't have a clear grasp of how numbers work, or we don't understand how human minds work, or both.
When we can see why infinities are impossible, I feel we will have learned something important about arithmetic, and perhaps about ourselves.

 

[Uncayimmy]

When I see these discussions I always wonder if I think too much or not enough. As I understand it, infinite simply means unbounded. I can relate to that considering how my good looks are unbounded.

 

[Ambrosia]

What gets annoying is when you take something finite, like a ball, and divide it up into infinitely many pieces. Then you spin the pieces, put the pieces back together, and you have two balls the same size as the original.

hmmm.... I even went and read the comic book version, and it didn't help either.

 

[Giraffe107]

I always thought that:
infinity-infinity=0
infinity/infinity=1

Also with the straight line thing I thought that was because the universe is... contained, for want of a better work, so it's not a copy of the earth but the same earth.

Am I wrong?

 

[Ambrosia]

A straight line in which direction? Or are they saying we're surrounded by sphere of countless trillions of duplicate earths located exactly (2^10¹¹⁸)*10²⁶m away from us?

Thats what they are saying. 10²⁶m is apparently the size of the observable universe, and that number is vanishingly small compared to 2^10¹¹⁸ which if I recall correctly is the number of particles in the observable universe, presumably sub atomic particles.

if you program a computer attached to a random-number generator, and had it print-out one character at a time at approximately the same rate a monkey would be expected to type, and let it go on printing random characters ....

then to get the phrase "to be or not to b" (assuming 1 second per character typed at random) would take longer than the time the universe has already existed. Thats 17 characters and the complete works runs to 5million + characters.

Hell to do one sonnet at around 600 characters would take 1 monkey at 1 character a second er [counts on fingers ... ] about 9.5*10⁸¹³ years. Thats being kind to the monkey in question and not caring about Capital letters or punctuation.

I need to go and lie down again.

 

[Terry]

Unless I'm also confused, while both sets of numbers might be infinite, the first infinite is twice as big as the second infinite. (Not all values of infinite are the same.)=

You are confused. While there are different sizes of infinity, both those sets are the same size of infinity. You can demonstrate this by providing a one-to-one correspondence between the elements in each of the sets.

 

[Brian-M]

I may be confused, or downright wrong (not unusual), but I don't see the 1:1 correspondence between the numbers in the sets.

Each number in the second set has an exact match in the first set, but that still leaves half the numbers in the first set unmatched. Dilb's example of multiplying the first set by two isn't entirely convincing to me.

From my point of view, multiplying by two might increase the total value of the numbers in the set, but in effect halves the number of numbers in the set. (If that makes any sense to you.)

If I am wrong in this, I will persist in being wrong until someone convinces me of why I'm wrong. You don't learn very much if you just accept someone else's assertion that you're wrong without understanding why.

 

[Roboramma]

I may be confused, or downright wrong (not unusual), but I don't see the 1:1 correspondence between the numbers in the sets.

Each number in the second set has an exact match in the first set, but that still leaves half the numbers in the first set unmatched. Dilb's example of multiplying the first set by two isn't entirely convincing to me.

From my point of view, multiplying by two might increase the total value of the numbers in the set, but in effect halves the number of numbers in the set. (If that makes any sense to you.)

Take the set of all whole numbers: 1,2,3,4,5, etc.
Mulitiply every number in that set by 2 (2,4,6,8,10, etc.)
Which even number is not represented in the second set?
The second set is: 2(1),2(2),2(3),2(4),2(5), etc.
For y=2(x) what whole number x is not represented by one and only one y?

Given this, there is necessarily a one to one correspondence between the two sets.

 

[Roboramma]

On the subject of infinity, this is one of my favourites:
http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Basically, you take a hotel with an infinite number of rooms, all of which are occupied.

What do you do when a new guest arrives, wanting a room?
What do you do when an infinite number of new guests arrive, each wanting a room?
What do you do when an infinite number of trains arrives, each holding an infinite number of new guests, each of which wants are a room?

 

[Roboramma]

What I find strange about the concept of infinite numbers is that it is obviously inconsistent with reality, (there are no concrete examples of infinite quantities and logically cannot be, because if there exists one infinite quantity of anything then how can there be anything else?)

If there are infinite spaces to put things, there's no contradiction between there being an infinite quantity of one thing and also some quantity of something else.
And if there is an infinite quantity of one thing, then there already is necessarily an infinite number of places to put things.

When we can see why infinities are impossible, I feel we will have learned something important about arithmetic, and perhaps about ourselves.

This sounds like basically saying, "intuitively infinities seem impossible, thus when we find out why they are impossible we will learn something important."
Maybe, but then again maybe what seems intuitively obvious simply isn't true. Personally, I'll wait and see.

 

[Roboramma]

Thats what they are saying.

Actually, I think they're saying that somewhere on that line you'll find an identical earth, but it won't necessarily be at the end of the line, just somewhere along it's length. So, somewhere within that sphere, not just on it's surface, along every cord there is an identical earth.

10²⁶m is apparently the size of the observable universe, and that number is vanishingly small compared to 2^10¹¹⁸ which if I recall correctly is the number of particles in the observable universe, presumably sub atomic particles.

I don't think that's the number of particles in the universe: the number seems far, far too large.
I think the number of protons in the observable universe is something like 10⁸⁰, which is, well, mind bogglingly smaller than 2¹⁰¹¹⁸.

I believe that what they are saying is that if the universe is infinite and has the same average density throughout as we see in the obervable universe, then, just by chance, sometimes a planet identical to earth will have formed, and far far more rarely, that planet will have had a history identical to ours. When you examine the probabilities, you can then figure out who rare such an occurance is, and thus, how far you'd have to go looking to find it.
That number above seems to be their answer.

 

[69dodge]

there are no concrete examples of infinite quantities ...

Perhaps.

... and logically cannot be, because if there exists one infinite quantity of anything then how can there be anything else?

No.

For example, there's no logical reason why there couldn't be infinitely many stars, and also lots of space between them for other stuff.

Just because there's space between the stars, therefore there are only finitely many of them? I don't see the connection at all.

I wonder if we mean the same thing by "infinite".

When we can see why infinities are impossible,

I don't think infinities are impossible.

 

[JoeTheJuggler]

I'm not sure what the connection is between number theory and the topology of the universe.

One is in the realm of mathematics, and the other physics.

While physics makes use of math and mathematical models, the definition of "infinity" doesn't say anything about the shape of the universe.

 

[69dodge]

I may be confused, or downright wrong (not unusual), but I don't see the 1:1 correspondence between the numbers in the sets.

Each number in the second set has an exact match in the first set, but that still leaves half the numbers in the first set unmatched.

It's important to realize that there are many possible correspondences between the two sets. Some of them are 1-1; some of them aren't. In order for the sets to be considered the same size, all you need is a single correspondence that's 1-1.

If you match each number in the even-only set with the same number in the other set, all the odd numbers will be left out. True. So don't do that. Match them up in such a way than none is left out. Provided this is possible, the two sets have the same size. And, in fact, it is possible: for example, you can match each number in the even-only set with the number in the other set that's half of it. Then, no number in either set is left out.

Dilb's example of multiplying the first set by two isn't entirely convincing to me.

From my point of view, multiplying by two might increase the total value of the numbers in the set, but in effect halves the number of numbers in the set.

Multiplying every number by 2 changes each number to a different one, but it doesn't change how many there are. For each one that you started with, you end up with exactly one, namely, its double.

 

[Roboramma]

I'm not sure what the connection is between number theory and the topology of the universe.

One is in the realm of mathematics, and the other physics.

While physics makes use of math and mathematical models, the definition of "infinity" doesn't say anything about the shape of the universe.

I don't think the reference in the OP was talking about the topology of the universe, though some people mistook it as such. The point isn't that if you go far enough you'll end up back where you started, it's that if you go far enough you'll end up in a different place that's exactly like the one you started from. And that follows if the universe is infinite, which is why it came up in this thread.