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**soha****Real Member**- Registered: 2006-07-07
- Posts: 2,530

A very large mathematical convention was held in Las Vegas. The conventioneers filled two hotels, each with an infinite number of rooms. The hotels were across the street from each other and were owned by brothers. One evening, while everyone was out at a bar-b-que, one of the hotels burned to the ground. The brothers got together and worked out a plan. In the remaining hotel, they moved all guests to twice their room number -- room 101 moved to 202, room 1234 moved to room 2468, etc. Then all the odd number rooms were empty, and there were an infinite number of odd rooms. So the guests from the other hotel moved into them

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,608

So, you can fit two infinities into one!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**RealEstateBroker****Member**- Registered: 2006-06-01
- Posts: 45

Some infinities are larger than other infinities.

If you have an infinite number of hands, then the number of fingers on those hands is also infinite, but the finger infinity is five times as great as the hand infinity.

When I was a kid I thought about infinity and other stuff, like does the Universe come to an end, and if there is a wall at the end of the Universe, how thick is the wall and what is on the other side. I Thought of infinity as an endless trains. You are standing at a train crossing, and there is one care after another going by and they just keep going on forever. You never get to see the caboose.

Then I thought that this train had a beginning, but no end. Suppose, instead, the train never began, and never ended? It had just been going by forever, and will continue to go on forever. That made me think that there are two types of infinity, which I called "one-tail infinity," and "two-tail infinity."

I wonder, though, if there actually exists any type of population with an infinite number of members? If there is nothing in existence which is actually infinite, then perhaps there really is no such thing as infinity, other than as a concept which we have created with our minds, which would explain how come there can be such paradoxes as infinities which are greater than other infinities.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

Some infinities are larger than other infinities.

If you have an infinite number of hands, then the number of fingers on those hands is also infinite, but the finger infinity is five times as great as the hand infinity.

Actually no. The problem in the opening post and your fingers and hands have nothing to do with larger infinities, but rather infinities of the same size.

There exists a bijection from the odd integers to the integers. All this means is that they are the same size. It is a fundamental theorem (axiom? Not sure) that two sets with a bijection between are the same size.

Because the even integers and the odd integers are both the same size as all the integers, we can "fit" the even and odd integers (both infinities) "into" all the integers (one infinity).

The same exact principle goes with your hand.

I wonder, though, if there actually exists any type of population with an infinite number of members? If there is nothing in existence which is actually infinite, then perhaps there really is no such thing as infinity, other than as a concept which we have created with our minds, which would explain how come there can be such paradoxes as infinities which are greater than other infinities.

What we find is that even though some mathimatical concepts don't actually exist in the real world, they still have applications in the real world. It's as true as it is weird.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,608

A bijection from integers to odd integers would be f(x) = 2x-1 (as an example, not the only one)

So for every integer there is an odd integer.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 20,794

For a similar discussion, go to Hilbert's hotel.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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