Okay, now we are converging. I spelled this out in my first comment but was sometimes lazy later on and did not mention it every time.
[...] then the resolution can not [only] be that you can not indefinitly divide space intervals in half.
So your position is essentially the paradox is resolved in our universe because the unstated assumption about the divisibility of space is not satisfied, but even if it was satisfied, Xeno's reasoning would still be wrong. So there is some more fundamental flaw in the reasoning but it doesn't really matter to you because you have a resolution for our universe by denying one of the assumption.
Strictly speaking you don't even know whether space and time are not indefinitely divisible in our universe, there are good reasons to think so, but it is - as far as I know - not a certainty. There are even good reasons to think that space does not exist in the same sense as temperature does not fundamentally exist and is something emerging from something more fundamental.
But whether or not our space and time are indefinitely divisible, just denying this assumption seems like the cheap way out for me, sidestepping the more fundamental issue that applies even if space and time are indefinitely divisible. So let me offer you a different view that does not hinge on the divisibility of space and time.
Xeno says, to reach the goal you first have to move half the way. Fine, I can do that. Then he says but in order to do that, you first have to move quarter of the way. Fine, I can do that. Then he says but in order to do that, you first have to move one eight of the way. Fine, I can do that. Then he says in order...and I get annoyed. Just tell me the first thing I have to do so we can start actually doing something. And now Xeno is in trouble, it is not that you can not do what he asks you to do, the problem is that Xeno can not make up his mind what you should do first.
Xeno can pick any arbitrarily large number, subdivide that often, and give you a list of things to do and you can easily do them, no matter how large a number he picked. You could even say, hey Xeno, let me just move to the goal. Does doing so include the task of going half of the way? Yes. Does this include the task of going quarter of the way? Yes. An eight? Yes? Is there any task in your infinite list that I did not complete? No.
The reasoning in the other version is more or less the same, you go half the way, then half the way of the rest, half the way of the rest, ... This time you seem stuck because Xeno can not make up his mind what the last task is you have to do, but again you can do all of them. This version has the nice property that you can actually start doing something as Xeno can at least tell you what the first task is, so you move half the way. And then another quarter. Another eight.
And while it might seem as if you are stuck as Xeno keeps throwing new tasks at you, you are not. We just have this tendency to think of tasks as requiring some fixed time and doing infinitely many of them will take forever. But if you keep track of time as Xeno throws more and more tasks at you, you will notice that time grinds to a halt. You complete one task after another but as you move with constant velocity each one takes less and less time and the total time approaches some limit slower and slower. So it takes you forever when measured in number of tasks but only something finite when measured in time. And again, just walking to the goal will include any task that Xeno would ever have asked you to do, so he can do something better with his time than trying to write a complete list of tasks for you.
And BTW, Newtonian mechanics are wrong, in the sense that they do not correspond to reality. (That doesn't mean they aren't useful.)
Wrong is the wrong term, it is an approximation. Take special relativity, in the limit of all velocities going to zero, you recover Newtonian mechanics exactly. Take general relativity, in the limit of slow motions, small masses, and low energies, you recover Newtonian mechanics exactly. The same for quantum mechanics. So if you approximately satisfy the assumptions, you will get a result with a very small error.
EDIT: I just thought of a variation of the problem, assuming indefinite divisibility of time. Xeno asks you to be in a certain place at 12:00. And 11:30. And 11:15. And 11:07:30. And so on. Admittedly being in a place is somewhat different than moving to somewhere, but would we call this a paradox? If we just stay at this place from 11:00 to 12:00 we can satisfy all requests Xeno makes, we can be there at all the times from any arbitrary finite or infinite list of times between 11:00 and 12:00. The list does not even have to be limited to countably many tasks as in the original paradox, it could contain uncountable many times, say the entire time between 11:11 and 11:12.
> Strictly speaking you don't even know whether space and time are not indefinitely divisible in our universe, there are good reasons to think so, but it is - as far as I know - not a certainty. There are even good reasons to think that space does not exist in the same sense as temperature does not fundamentally exist and is something emerging from something more fundamental.
Yes, I agree, our knowledge is imperfect and the truth may be different.
> But whether or not our space and time are indefinitely divisible, just denying this assumption seems like the cheap way out for me, sidestepping the more fundamental issue that applies even if space and time are indefinitely divisible.
Here we disagree. I think the whole point of the exercise from Zeno's point-of-view is to make you question your assumptions.
> So let me offer you a different view that does not hinge on the divisibility of space and time.
I didn't find your resolution very satisfying - it seemed to boil down to "motion is clearly possible by demonstration" and "you can't make an infinite list". Both of these things are part of Zeno's argument (the second in a different form).
> And again, just walking to the goal will include any task that Xeno would ever have asked you to do
When you get down to the quantum scale, is this still true? Does it even make sense to talk about a human moving a very small fraction of the width of an atom?
> The list does not even have to be limited to countably many tasks as in the original paradox, it could contain uncountable many times, say the entire time between 11:11 and 11:12.
Would it surprise you to learn that I think the universe is at most countable, if not finite?
Here we disagree. I think the whole point of the exercise from Zeno's point-of-view is to make you question your assumptions.
Sure, but we also agree that it would be weird if it was impossible to move in Euclidean space, so I would say further questioning is warranted, what other assumptions are wrong that also resolve the paradox for Euclidean space.
I didn't find your resolution very satisfying - it seemed to boil down to "motion is clearly possible by demonstration" and "you can't make an infinite list". Both of these things are part of Zeno's argument (the second in a different form).
I would say the core of my argument is that Xeno essentially argues that you can not do an infinite number of things in finite time at that this is a wrong assumption. Look, I can not even finish telling you all the things you have to do, how could you possibly do them all?
And as I said, just being in one place from 11:00 to 12:00 makes you do being at that place at all times between 11:00 to 12:00, independent of whether there are finitely many, countably many, or uncountably many instants in that interval. It is also noteworthy that doing something here is a pretty weak form of doing something, being at a place at some instant does not take any time at all or the smallest possible amount of time if time is not continuous. But similarly is moving infinitesimal distances in the original paradox a pretty weak form of doing something.
But now that I think about it again, my argument can not replace your argument as I rely on the indefinit divisibility. So what I would really want to say is probably either space and time or not indefinitely divisibel, then Xeno can not make an infinite list of tasks to do, or space and time are indefinitely divisibel, but then it is also possible to do infinitely many tasks in finite time notwithstanding Xeno's inability to enumerate them.
Would it surprise you to learn that I think the universe is at most countable, if not finite?
Not really, I am also sympathetic towards ideas like loop quantum gravity. I also always liked - for no good reason - the idea of a closed universe but the observed flatness seems to point into a different direction. On the other hand someone pointed out to me some time ago that there are closed flat topologies, for example a torus, which I knew but never considered when thinking about the universe.