There seems to be a contradiction with using numbers and multiplication to create an area

[ryan]

Assume a universe capable of having in it a continuous area of, say, 3 meters squared. An infinite number of points exist in this area. For example (2,1) where x = 2 and y = 1 on a Cartesian plane is one of many, but its area is 0. So it seems to imply an infinite number of 0 spaces makes a space.

However, if we use an epsilon between a converging function to a limit, we will end up with the right answer 3. (to see how this is done watch the video showing how infinity*0 can equal 1, https://www.youtube.com/watch?v=bSZiEZzwB7Q )

But in the definition of a limit, epsilon can't equal zero.

Am I missing something here?

 

[Speakpigeon]

Seems like the paradox of Achilles and the tortoise all over again.

The area of one point is zero, the area of any finite number of points, including any very large but finite number of points is zero, and the area of any infinite but countable number of points is zero, at least as far as I see it. Yes?

Yet, a m2 is a m2

So, apparently we just move from 0 to 3 m2 just by moving from any countable number of points to any R-like infinity of points. But where would be the problem exactly? OK, there's an intuitive problem, surely, since mathematicians seem to have struggled a bit to get it right and that there are still echoes of this today. But, me, I ca'tn see any logical issue. As I would say myself, this is an empirical question and it has been solved, apparently to everybody's satisfaction.

So, perhaps, you may want to ask yourself why it is that the area of 2 points is zero. Because, as far as I understand, the notion of area is never conceived of as that of any finite number of points, except for saying that it is zero anyway. Seems a bit like talking about zero elephant. One elephant, yes. Zero elephant, that's not like an elephant that wouldn't be there, although this seems to be in effect how our mind think of it. So, I would say that our idea of any exiting area sort of pops out into existence providing we're thinking of an actual R-like infinity of points arranged in two dimensions. Otherwise it's zero elephants. And if it's a countable infinity of points, then the area is like zero elephants all the way down.

Think also that there is no continuity between finite sets and infinite sets, and again no continuum between Q and R. And if there's no continuity, we can't argue that the area of any countably finite number of points should be anything else than zero. It is zero not because we add up an infinity of zeroes as you suggests, but simply because the notion of surface just doesn't make any sense in this case.

There, you have your nice little emergent property, surface. An area is the measure of a surface. A countably infinite number of points just doesn't make it a surface. So, zero.
EB

 

[ryan]

Seems like the paradox of Achilles and the tortoise all over again.

The area of one point is zero, the area of any finite number of points, including any very large but finite number of points is zero, and the area of any infinite but countable number of points is zero, at least as far as I see it. Yes?

I don't think so. Riemann sums use natural numbers (when n goes infinity) for integration to form areas. It seems that a countable infinity is sufficient to create areas.