[bonanova]

Infinity gives rise to lots of counter-intuitive results - let's call them paradoxes; here are two.

[1] Two times infinity equals infinity.

Take the set of positive integers 1, 2, 3, 4, ...

One might think that's it - there aren't any integers other then these.

Ah, but there are: multiply them by 2 to get 2, 4, 6, 8, .... you can assign an integer to each even integer.

But that leaves "holes" in the sequence at 1, 3, 5, 7, ... and you can assign an integer to each of these holes.

So without adding any integers to the original set, you have doubled, or replicated the entire set.

This is another way of saying that the cardinality of integers, even integers and odd integers is the same.

I = E + O. But I, E and O are all the same size! Certainly a strange way of thinking about "size" of sets.

[2] Banach-Tarski Paradox

The second one is breathtaking - it bears the names of two mathematicians, Stefan Banach and Alfred Tarski.

In 1924 they wrote a paper which describes how the infinite set of points {x,y,z} | x²+y²+z²<=1 can be divided

into no more than 5 partitions - one of them is the point 0,0,0 - which are then translated and rotated, but not

stretched, so their shape is not changed, and re-assembled into two separate sets of points, each identical to

the first! Or into a sphere of larger radius than the original. As one person put it, "A pea can be split into a finite

number of pieces and be re-assembled into a sphere the size of the Sun."

If you follow the duplication of positive integers in [1] you can kind of appreciate how this might come about.

But it's beyond simply renumbering the initial set of points. In fact, you can't do it in 1 or 2 dimensions at all.

The five pieces are infinitely complex - to the extent that they are not "measurable". It's impossible to determine

their volume. So when they're re-assembled, the final volume can be different from the original volume.

Again, intuition is thrown out the window.

BTP relies on something that most mathematicians accept [but not all agree with] called the Axiom of Choice.

AOC states that if you have a collection of sets - perhaps an uncountably large collection - there exists a set

called the Choice Set that contains exactly one element of each of the original sets. That is, without saying precisely how an

element from each set is to be chosen, it can be done, and you can talk about a representative member of

each of the sets. This is equivalent, actually, to the well-ordering theorem, which in effect says you can find

and then choose the "least" or "smallest" member of any set.

If AOC is removed, then partitions are always measurable, and any duplication or expansion of volume is not

possible. Not surprisingly, non-measurable sets have provided a strong argument against AOC. But AOC has the

virtue of providing easy proofs of important theorems. So it's generally accepted, and imponderables like the

BTP remain with us.

So that's it - if you mess with infinity, you may need to park your intuition at the door.

[Guest]

Ahh! I don't like infinity.

It's too complicated for me.

I don't get the pea thingie. How can it be like that?

It's rather like magic, Like a mix between mathematics and magic.

[Guest]

Sorry about the double post. I was going to just sit round and be lazy as I figured it would be too hard to solve. But I changed my mind. Here's the lowest answer I've gotten so far:

7 Days (1 week)

To get this, I drew and 11x11 grid with the length representing the viewing groups while the height represented the performing groups. I then proceeded to try and take the biggest chunks of the grid out every day. I also figure that if there were twelve groups, they could still get through in seven days as well.

Day 1:

1, 2, 3, 4, 5 are watched by 6, 7, 8, 9, 10, 11

Day 2:

6, 7, 8, 9, 10, 11 are watched by 1, 2, 3, 4, 5

Day 3:

1, 2 are watched by 3, 4, 5

6, 7, 8 are watched by 9, 10, 11

Day 4:

3, 4, 5 are watched by 1, 2

9, 10, 11 are watched by 6, 7, 8

Day 5:

1 is watched by 2

3 is watched by 4, 5

6 is watched by 7, 8

9 is watched by 10, 11

Day 6:

2 is watched by 1

4 is watched by 3, 5

7 is watched by 6, 8

10 is watched by 9, 11

Day 7:

5 is watched by 3, 4

8 is watched by 6, 7

11 is watched by 9, 10

This is the best I can come up with. In the later days, groups would be able to watch groups they've already seen of course. Eg, by day 7, groups 1 and 2 have seen every performance and been seen by every performance so they may as well go home.

This answer doesn't feel quite as efficient as it could be so I figure someone might be able to find an answer for 6 days but that would be as low as it would go.

Alex and his friends Ian and Davey were enjoying

some cold ones at Morty's last night when Alex pulled

a small pouch out of his pocket and began to tell his

story.

So this fellow I met claimed he mastered the art of

loading dice, and he was able to separately adjust the

chances of each of the faces showing. We got to talking,

and I asked him to make a pair of dice that would make

all the possible results, 2, 3, 4, ... 10, 11 and 12,

appear with equal chances. Here they are. And

he held them up for all to see.

No way, said Davey, who was no mathematician, to

be sure, but this seemed beyond possibility.

So you're an unbeliever, replied Alex. Then watch this;

and Alex rolled the dice 5 times, getting 2, 5 12, 9 and 6.

How many times would you have to roll an honest pair

of dice to see snake eyes and box cars in 5 rolls?

Jamie heard the commotion and strolled over to the table.

Here, let me try. And Jamie rolled 3, 10, 8, 2 and 11.

Well, these are no normal dice, that's for sure, he said.

What are you going to do with them?

Haven't decided yet, Alex replied. For now, maybe just

a few little wagers with some of my, ahem, acquaintances.

Ian had been thinking for a bit. They're special dice,

to be sure, he said. But I'm quite certain they're not

what you say they are.

If you were there, who would you agree with?

It can also be 2x infinity correct ? and 4x, 5x, ect. . ?

[Guest]

Edit: [etc...]

BTP relies on something that most mathematicians accept [but not all agree with] called the Axiom of Choice.

AOC states that if you have a collection of sets - perhaps an uncountably large collection - there exists a set

called the Choice Set that contains exactly one element of each. That is, without saying precisely how an

element from each set is to be chosen, it can be done, and you can talk about a representative member of

each of the sets. This is equivalent, actually, to the well-ordering theorem, which in effect says you can find

and then choose the "least" or "smallest" member of any set.

If AOC is removed, then partitions are always measurable, and any duplication or expansion of volume is not

possible. Not surprisingly, non-measurable sets have provided a strong argument against AOC. But it has the

virtue of providing easy proofs of important theorems. So it's generally accepted, and imponderables like the

BTP remain with us.

One of the world's nerdiest quotes:

"The Axiom of Choice is obviously true, the Well Ordering Principle obviously false, and who can tell about Zorn's Lemma?" -- Jerry Bona

Edited December 22, 2008 by d3k3

[Guest]

another one with infinity:

The Property of infinity is infinity time anything is infinity.

The Property of zero is zero times anything is zero.

So Whats Zero times Infinity?

[Guest]

Zero. Zero is more powerful than infinity. Emptiness is more powerful than matter. Etcetera.

I think that's an interesting concept for a fictional villain in a fantasy setting.

[Guest]

Zero times infinity is indeterminate. Some zeros are more powerful than infinity some aren't. Is that paradoxical? or mathematical?

[Guest]

another one with infinity:

The Property of infinity is infinity time anything is infinity.

The Property of zero is zero times anything is zero.

So Whats Zero times Infinity?

the nth dimension without any contents.

The question I ask is this - is limitless empty space more than simply "nothing" ?

Is a graph without any data still a graph?

To me these forms of nothingness are a prerequisite for something. Or, if you can't have nothing, you can't have something.

[Guest]

Anything, positive integers, or negative integers, multiplied by zero, is zero. It's like a delete button. If you were to multiply negative 1 by positive infinity, you would get negative infinity, because infinity is still an integer, no matter how you look at it. Zero overpowers integers.

If a graph has no data, then it's still a graph - it just has no data on it. Pick up a pencil and write some bars, you lazy a**.

Limitless empty space is truly nothing more than nothing. Until you put something into the empty space. Then it becomes a storage area.

If a pea were sliced into impossibly thin slices, every molecule separated, perhaps it could be the size of the sun. But you'd need a pea the size of the earth or larger to begin with for humans to ever be able to do it.

That's all I have to say on the subject.

[Guest]

stuff...because infinity is still an integer,...stuff

Nope. Not an integer.

[Cavenglok]

...what?

[Guest]

So...what are the arguments for/against zero being an integer or not? I always thought it WAS an integer...

[Guest]

Infinity is not a number. Zero is a number. If we multiply infinity by zero it's undefined. However, this is where the concept of limits goes. Any number close to infinity multiplied by 0 is zero.

[Guest]

zero is nothingness and nothing is more powerful then nothingness. for something that actually IS NOT, nothingness is very powerful, now that's a paradox

[bonanova]

So...what are the arguments for/against zero being an integer or not? I always thought it WAS an integer...

Every integer has a next-highest integer.

[bonanova]

zero is nothingness and nothing is more powerful then nothingness. for something that actually IS NOT, nothingness is very powerful, now that's a paradox

Take the quantities 1/x² and x. They have limits of infinity and zero as x goes to zero. What about their product?

(1/x²) x (x) = 1/x. The limit is infinity.

Change the quantities: (1/x) x (x²) = x. The limit is zero.

Change the quantities again: (1/x) x (x) = 1. The limit is unity.

[Guest]

to : bonanova

how do you perceive zero?

to : Matt S.

http://en.wikipedia.org/wiki/Integer

[Guest]

another one with infinity:

The Property of infinity is infinity time anything is infinity.

The Property of zero is zero times anything is zero.

So Whats Zero times Infinity?

Infinity is not really a number, but let's do this as if it was...

First, let's explain those sentences visually. (・・・ mean repeating this pattern)

The Property of infinity is infinity time anything is infinity. ∞ * x = ∞

The Property of zero is zero times anything is zero. 0 * x = 0

So Whats Zero times Infinity? 0 * ∞ = y or ??? haha

if x = ∞ then this is simple.

∞ * ∞ = ∞ statement 1 is true.

∞ * 0 = 0 is this true? let's find out...

∞ = ∞ * x where x can be any number,

so:

∞(2) = ∞

∞ * 3²⁷ = ∞

and so on...

∞ * -1 = -∞ (uh oh, haha just kidding, ignore that one )

anyway, let's continue...

so:

2^∞ = ∞ (two to the infinite power equals infinity)

2(2*2*2*2*・・・) = 2^∞ = ∞ (two multiplied by two times two times two times an infinite number of two times two is equal to two times the infinite power which is equal to infinity)

so:

2(2*2*2*2*・・・) = ∞. (two multiplied by two times two times two times an infinite number of two times two is equal to infinity)

now...

2(2*2*2*2*・・・) * 0 = 0 (two multiplied by two times two times two times an infinite number of two times two multiplied by zero equals zero)

so transitively speaking,

if

2(2*2*2*2*・・・) = ∞ (two multiplied by two times two times two times an infinite number of two times two equals infinity)

and

2(2*2*2*2*・・・) * 0 = 0 (two multiplied by two times two times two times an infinite number of two times two multiplied by zero equals zero)

then

∞ * 0 = 0 (infinity times zero equals zero)

simple math

[Guest]

another one with infinity:

The Property of infinity is infinity time anything is infinity.

The Property of zero is zero times anything is zero.

So Whats Zero times Infinity?

(sorry i dont know how to quote)

Well, if you had infinity zero times, then you would have nothing. If you had zero infinity times, then you would have a bunch of 0s, which is nothing. 0 + 0 + 0 + 0 + 0 + 0... equals 0.

[bonanova]

to : bonanova

how do you perceive zero?

to : Matt S.

http://en.wikipedia.org/wiki/Integer

Thanks for asking!

First, zero is the identity element for addition.

But there's