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Budimir Zdravkovic

PhD student in biochemistry/cancer biology,

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Can three infinities be greater than one?

The idea is very simple but the conclusion may redefine our very concept of infinity and quantity unless I am wrong.

There are infinite possible positions in one dimensional space. That is if we assume that a given position is infinitely small or that one dimensonal space stretches infinitely. Two dimensonal space offers more possibilities than one dimensonal space since two dimensional space has all possible positions of one dimensional space plus all possible positions that can occupy the second dimenson that was added. And we can keep adding dimensions thus creating more possible positions with each added infinity.

What seems to emerge is that as you add more dimensons you are adding more possible positions relative to the previous number of dimensons but you are not necessarily increasing the number of positions because the number will always be infinity. Thus you can add more to a quantity without increasing it.

Weird.

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  • Mar 31 2011: Infinite is infinite, but the absolute distance between two random points in an infinite 1d, 2d, and 3d space (which have equal point densities on their axises) would tend to be smaller as you increase dimensions. So while an infinite 1d space provides an equally unending number of locations as an infinite 3d space, the 3d space has more locations that are of near proximity than the 1d space. As you described them, these would each represent different densities in a given infinite space. But simply putting the points closer together would have the same affect. An infinite 1d line where the points where 1cm apart would be less dense than an infinite line where the points were 1mm apart; but both would have an equally infinite number of points.

    Infinite can be different in rates of growth as well as densities. A line y = x grows at a rate of O(n) as it approaches infinity, but a curve y = x^2 grows at a rate of O(N^2) or exponentially faster as it approaches infinity. Rates of growth as you approach infinite (represented here by Big O Notation*) are important and they demonstrate how two different infinite quantities can behave differently. But, in the end, they both contain the same unending quantity of whatever.

    Big O Notaton: http://en.wikipedia.org/wiki/Big_O_notation
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      Apr 1 2011: what is the absolute distance between two random points? give me a number
      • Apr 1 2011: The absolute distance would be whatever straight line distance between the two points exists (after looking this up the term I was looking for was Euclidean distance). But when choosing random points in a more densely populated space, you're more likely to choose points closer together than two randomly chosen points in a less densely populated space. It's just a probability demonstration of how two infinite sets can behave differently even though both have the same unending quantity of points.

        My math knowledge is somewhat limited to what I've specialized in. As a programmer, I do work with growth rates as quantities approach infinite daily. I don't remember how to make my example clear as a mathematical representation, but obviously there's no one number that represents the distance between two random points. But there would be equations that demonstrate the probable difference when selecting from the two different pools of points.
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          Apr 1 2011: i didn't mean to ask for definition, but an example. my point is that your statements are ill-defined. i'm not a mathematician either, so i don't know if we can talk about evenly distributed random number from an infinite set. if there is such a thing, then the average distance between to random point on the line is infinite. similarly, the distance of two random points in space is also infinite. so you can't really say one is bigger than the other. or can you?
      • Apr 4 2011: @Krisztián Pintér Ya, my math skills aren't what they used to be. Thinking about what you said, I think you're right. As you approach infinity when picking random points, the distance between the two points would also approach infinite no matter what the density of the points would be. I wish I had time to write the proof to figure it out (or even look it up) but I don't. If that is the case, though, it would certainly be further evidence that the different infinites are still equally infinite even though one is more densely populated than the other.

        This is making me wish I was back in college as a Math major. I forgot how much fun it can be.
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      Apr 2 2011: I understand that with additional dimensons each point is surrounded by more "neighbouring" points. But in terms of the distance decreasing, why does a given distance decrease, is it something to do with indtroducing new vectors into the space?

      I only had to take two years of calculus in college and that was also a long time ago so I apologize if my knowledge is rusty. In what I do I only need roughly 10% of the calculus I learned. .
      • Apr 4 2011: Well as Krisztián pointed out, the distances actually approach infinite as the random selections approach infinite, but I was thinking more within a given range (which was where I made my mistake). But within that finite space, if you plot out 1 million points an equal distance apart from each other in both a 1d and 3d space, the distance between the two furthest points on the 1d line would be 1 million units. Where in the 3d space, you would only have 100 units of length on each side of the cube (to create a cube with 1 million 1 unit spaced points within it). So the distance between the points furthest from each other would be only 100 units.

        A visual representation of a smaller scale could be thought of as a piece of paper with 64 equally spaced points in a line for a 1d space, then a piece of paper with a 8x8 square of points for the 2d space, then for the 3d space, you need a 4x4x4 cube of points. So when you measure the furthest straight line distance between the points in the different spaces, you get 64, 8 and 4 respectively. It's kind of like how the more you fold a piece of paper the "smaller" it gets.

        But it was all wrong to show differences in infinite groups because as you approach infinity picking random points also creates a distance between those points that approaches infinity. I think it would actually prove the opposite, in that the different densities of infinite spaces would actually behave the same, in my example, as you approach infinity.

        The order of growth stuff was correct though. It's helpful in programming so we can figure out if an algorithm is useful or not for very large inputs. If the order of growth is too great, when inputs are very large it would simply take too long, or too much memory, to use that algorithm.

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