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Hey,

Quote from: Colin Michael on June 06, 2009, 10:32:44 PM

Infinity x X or Y is still infinity. I'd say that's an equality.

Even when fudging the math and considering the idea of infinity in equations, the equation "infinity = infinity" is not always true.  There are different orders of infinity that are not "equal" to each other.  The way it was explained to me was that if hotel A has one floor and an infinite number of rooms on each floor and hotel B has an infinite number of floors and an infinite number of rooms on each floor then both hotels have an infinite number of rooms but hotel B has far more rooms than hotel A.

Tschow,

Tim "Sir Nobody" Maly

I think what you're trying to get at here is "countably infinite" versus "uncountably infinite."

A set is finite if and and only if its elements can be mapped to {1, 2, 3, 4,...X} where X > 0 and is an integer.
A set is countably infinite if and if only if its elements can be mapped to the set I = {1, 2, 3, 4,...}
If a set is neither countably infinite nor finite, then it is uncountably infinite.

In your example here, even if hotel B contained an infinite number of floors with an infinite number of rooms, it is still "only" countably infinite because I can map each floor to the set {1,2,3,...} and on each floor, I can map each room to the set {1,2,3,4...}. The union of two countably infinite sets is still countably infinite. In the vernacular, there is no mathematical difference between hotel A and hotel B.

An example of an uncountably infinite set is the set of real numbers R=[0,1]. If you try to set up a correspondence to the set {1,2,3,...} say by letting x2 >x1, x1 in [0,1] -> 1, x2 in [0,1] -> 2, etc., then to what element in I does (x1+x2)/2 correspond? This is the problem. No matter how small the difference between x2 and x1, you can always find an element (x1+x2)/2 that will not be able to be mapped to an element in I. You can always "squeeze another number in." Since the set is clearly not finite, and a one-to-one mapping between R and I cannot be established, R is uncountably infinite.

Another example of an uncountably infinite set is the Cantor set, K. This is obtained by deleting the middle third (1/3, 2/3) from [0,1]. Then, you delete the middle third from the segments [0,1/3) and (2/3,1]. Repeat this process ad infinitum, and the dust of points that remain is K. K is uncountably infinite for reasons that are beyond the scope of this post. :-)

-Egyptian

Quote from: Egyptian on June 07, 2009, 01:52:21 PM

Another example of an uncountably infinite set is the Cantor set, K. This is obtained by deleting the middle third (1/3, 2/3) from [0,1]. Then, you delete the middle third from the segments [0,1/3) and (2/3,1]. Repeat this process ad infinitum, and the dust of points that remain is K. K is uncountably infinite for reasons that are beyond the scope of this post. :-)

This is Chaos!  

What I love about K is that it "seems" infinitely "less" than [0,1] yet is itself uncountably infinite!

Hey,

I think what you're trying to get at here is "countably infinite" versus "uncountably infinite."

Right!  That...wow my math is rusty.  The uncountability of the real numbers never sat well with me.

Tschow,

Tim "Sir Nobody" Maly

I'm kind of confused and lost with all of this.

You're confused by big numbers and I'm confused by big words. I say we're even.  

The uncountability of the real numbers never sat well with me.

There are support groups for that, you know. The first step is admitting you have a problem... LOL