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GermanStar - 5/13/2005 6:23 PM
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Botnst - 5/13/2005 2:51 PM
No over here I have a bag with an infinite number of integers. I am now going to label each line that intersects each point on the first line with each of my integers. I pull out an integer and stick it on a line and go to the next line. each labeled line is discrete just like each integer in my bag. I will use up all my integers labeling every line that intersects that very first point within just this one plane. When I finally get all of them enumerated, I have to move to the next point with it's infinite set of intersecting lines, but I have no integers left in my bag after having enumerated the first infinite set and I have an infinite set of sets that still need labels!
I believe I understand your point, but your argument falls apart by considering the discrepancy in the two phrases I've highlighted. It is not possible to run out of an infinite supply, so your argument renders the number of integers as finite, rather than infinite.
Well, that's a problem isn't it? Try this one to see if it provides a useful tool.
A line extends infinitely in two directions. If I cut it anywhere, it is exactly at the midpoint.
Huh?
If I cut the line anywhere, how far will it go to the left on the left half, and how far to the right on the right half? Infinitely for both, correct? So the two portions are the same length since both extend to infinity.
Once you start geting the rhythm of this thing its kind of interesting.
There are also descriptions of infinities that are interesting. For example, the number of points on the surface of a plane is infinite and unbounded. That is, the plane surface has infinite points and it extends in every direction and never folds back on itself. In comparison, the surface of a sphere also has infinite points but since it is curved and folds up on itself, it is bounded.
can you imagine a curved surface that is both infinite and unbounded? A paraboloid or hyperboloid, etc.
How about something that curves through all three dimensions but in different directions that change with each point, but never folds back on itself?
each of these family of shapes represent different descriptions of infinites bounded by two or three dimensions. One could map points from one onto points on the other.
But what if the number of dimensions were infinite?