Thinking about infinity

[robvalue]

Yes, you're correct, it can be to do with Zeno's paradox.

It's the problem of insisting we only take finitely many steps and thus never reach our target. But (abstractly at least) we take infinitely many "steps"; each of smaller and smaller lengths.

In reality, it may or may not work like this. Is there a minimum distance something can move? If there is, then you can't take infinitely many steps. There is only a finite number you can take, no matter how small, between two points a finite distance apart. Your first step must be at least this amount (it would likely we would be dealing with multiples of this smallest amount I think). We would in effect be "jumping" from one point to another, without ever occupying the space inbetween.

But if there is no minimum amount, then you have moved through infinitely many. Disallowing this would, as you correctly observe, prevent you from even starting to move at all. But you are essentially stopping the "time taken" from ever reaching a certain point, which is not valid.

It's a weird concept to be sure. But there is no problem at all with a finite number being an infinite sum, abstractly. The only question is whether reality can be subdivided the same way. If there is no limit to how small I can cut something up, I could go on forever. Obviously, if I'm only allowed to make finitely many cuts, I can't make them all. But that's the essence of Zeno's paradox. Insisting on finitely many is the equivalent of insisting I cannot move in the first place. It's assuming the conclusion by disallowing infinities as a premise.

So, for example, let's say an object is of length 1 metre. Let's say there is no minimum length that a subdivision can be. Then, if we ask how many subdivisions of it are there if we use these lengths (I used this example before):

1/2 + 1/4 + 1/8 + ...

then the answer is infinitely many. It doesn't matter if I actually go in and mark them all, obviously that's impossible in a finite amount of time if it takes me a fixed amount of time for each cut. But if the time taken is proportional to the distance I have to move instead to make the cut, then I can theoretically make all the cuts.

It does consist of an infinite amount of them, whether or not they are actually "marked" by someone, or cut into pieces. However, if we reach a minimum distance we can cut, this no longer works. Were confined to (length/minimum distance) numbers of partitions.

Let X = 1/2 + 1/4 + 1/8 + ...

Then 2X = 1 + 1/2 + 1/4 + 1/8 + ...

2X - X = 1 [the rest of the terms all cancel out]

X = 1

This isn't a trick, it's a genuine mathematical method.