Infinities, part one

Infinity. It's a really big number—the biggest, maybe. Infinity plus 2 is still infinity. Infinity times 2 is also still infinity. Infinity seems to have no interaction with other numbers, except zero. What would infinity squared be? Infinity? Yes and no. It wouldn't be finite, so it seems it should be.

Imagine this: You have a line segment, which always has an infinite number of points. Since the distance from one endpoint to the other on a line segment is finite, the spacing between the points is zero, which is any finite number divided by infinity. The reason is the spacing between evenly spaced points on a line segment is the length of the line divided by the number of points on it; any finite number divided by infinity, or zero. Double it's length. The spacing should therefore be doubled, but is still zero, because two times zero is zero. Any other line segment also has zero spacing between points, because zero times any finite number is still zero. But if the segment became a line (stretching endlessly in both directions), the space between points along it should be infinitely times more. The limit as x approaches zero of the inverse of x equals infinity, so zero times infinity equals zero divided by zero For details about limits, check out http://www.youtube.com/watch?v=HYSI-AHUqRM. Therefore, if s is the spacing between points on a line, we get the following equation:

0/0 = s

and if we multiply both sides by zero we get

0 = 0s.

What times zero equals zero? The answer is anything. For the moment, lets pick three inches. But that seems impossible, there can't be three inches between all the possible points. If we multiply the number of points by 2, the space between points will be half, as in 1.5 inches between each point, and two times infinity is still infinity. Multiplying the number of points by any finite number will leave that number at infinity, and still impossible. Therefore, you can't have an infinity number of points on a line. You have to multiply the spacing between points by zero for it to be true, as in dividing the number of points by 0, or multiplying them by infinity. Therefore, the number of points on a line is infinity times infinity. This can't be infinity, because we proved that can't work. Another reason this makes sense is that infinity is infinitely times any finite number, which is why multiplying or adding infinity by any finite number has no effect on infinity. But infinity is the same as infinity, therefore it has an effect. Since there are an infinite number of lines in a strip that extends forever, and an infinite number of those in a plane, we see there are even more. And because we can mathematically think of an infinite number of dimensions, there have to be an infinite number of different types of infinities, or more. So what is infinity times infinity?

First, we need a new type of notation. Our conceptual infinity can be represented by aleph null (as shown below)

How would you represent infinity times infinity? it would be aleph sub 1, as shown.

What would Aleph sub 2 be? Not infinity times infinity times infinity, but would be that times infinity again. The reason? It's twice the amount of infinities as the one before. The reason behind that? It is therefore the previous one squared. Therefore infinity to the x power is aleph sub log base two of x. Think about it. Infinity to the first power is infinity which is aleph null. aleph sub log base two of 1 is aleph null. Infinity squared is aleph sub one, and aleph sub log base two of two is aleph sub one. you can convert an infinity in aleph sub n form to infinity to a power form as well. To do this, you have the equation aleph sub n equals infinity to the two to n power. For multiplying two infinities, un-simplify them into infinity to a power form, add the exponents, then re-simplify. For example, aleph sub two times aleph sub two should be aleph sub three because it's aleph sub two squared. You can put that into infinity to the fourth power times infinity to the fourth power. By adding the exponents, you get infinity to the eighth power, which is aleph sub log base two of eight. This turns out to be aleph sub three, as we predicted.

What about infinity to the aleph null power? That would be aleph sub log base two of infinity. Log base two of infinity is infinity, so it would be aleph sub infinity, or aleph sub aleph null. If you have aleph sub an infinite number, all the same rules apply to that base and to the base of the base. What is aleph sub aleph null? it is infinity times infinity aleph null times, as in the number of points in an object with aleph null dimensions. Aleph sub aleph sub one would be the last answer times itself aleph null times, or the number of points in an object with aleph sub one dimensions. Part two will include more rules and more to do with aleph sub infinite numbers.
for more on Aleph and the Aleph notation, check out Cantor on Youtube or Google video or check out his biography at http://www.youtube.com/view_play_list?p=1EEE2070E3685518