Actual Infinity in Reality?

[polymath257]

And if there are only finitely many points between, then there will be a last point. But if there are infinitely many, there does not have to be. That is simply one of the ways infinite sets differ from finite ones: infinite ones need not have a first or last.

The underlying assumption you seem to make is that an infinite number of steps cannot be finished. But they can, *if* they are done faster and faster, so that the total time is still finite.

So, the difficulty seems to be that an infinite sum of positive quantities can be finite. it is easy to see that they can be bounded. For example:

.9
.99
.999
.9999
.99999
...

I think you will agree that there are infinitely many numbers in this list (whether potential or actual). But, clearly, every number on this list is smaller than 2.

So this infinite set (potential, if you like) is bounded above by 2.

This is an example of an infinite, but bounded collection. Even if you only want to do 'potential' infinities, the numbers that appear are ALL smaller than 2 (and, in fact, each is smaller than 1).

So, even with a potential infinity, it is possible to be infinite and bounded. It is infinite in quantity, but finite in terms of size of the numbers involved.

And, if these represent time passage, we DO go through all of them and reach 1. Not by staying in this sequence, but by going through the sequence in a finite time.

You may do things as fast or slow as you like.   In Zeno's dichotomy paradox, and your X<Y<1    no matter how much time, you will never reach 1, you will always have another point to go through that is less than 1 (hence why the term infinity applies). This is not an assumption... this is the math function being described!  The argument is not that it will take an extraordinary amount of time to complete!

Let me do it like this. Suppose you are attempting to go through the sequence .9, .99, .999, .9999, .99999, .... There are two scenarios..

In the first, you are at .9 after 1 second, at .99 after 2 seconds, .999 after 3 seconds, etc. I agree that at this rate you will never get through the sequence: it will take an infinite amount of time to do so.

In the second, though, you are at .9 after 1 second, .99 after 1.5 seconds, .999 after 1.75 seconds, .9999 after 1.875 seconds, etc. Each step takes half the time of the previous one. In this scenario, you *will* get through the sequence of 9's after 2 seconds. At the 2 second mark you are past every single one of them.

And yes, whenever you are at any of those points, you have more to go through.
And yes, when you get to t=2, you are past all of them.
And no, there isn't a last one you went through.
And yes, you went through all of them.

The point: you do, in fact, go through an infinite number of times in every finite duration.