Zeno's Paradox as proof of a finite universe

Zeno's Paradox says (basically) that movement is not possible because in order to move any distance one must first move half that distance, and half that distance, etc etc. He assumes that the physical universe mirrors mathematics in that you can divide a distance infinitely just as you can divide a number infinitely. Simply said he assumes that the universe is infinitely large since no finite number can be divided infinitely. However, empirically Zeno's Paradox quite obviously doesn't hold any water.

So let's look at it backwards, the reason we can't move would be, indirectly, that the universe is infinitely large. Since we can move, the universe must not be infinitely large but rather be composed of a near infinite amount of fundamental units of mass.

I'm not suggesting this is some groundbreaking idea, but I wonder if anyone has reason to dispute it? Or if some philosopher has already stated it (Zeno may have, I'm not sure).

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Response to izubachi:
I'm familiar with the limit, but it doesn't really disprove what I'm saying. What we're talking about here is the limit at an asymptote However, although it will tell what number y will move towards as x goes to infinity (for a horizontal asymptote) and at what value of x, y will move towards infinity. It doesn't change the fact that the numbers never actually reach infinity or whatever number they were approaching. So if you were trying to move an infinite distance you would also never actually reach your destination. But I'm not a mathematician, so I may be mistaken.

As for relativity, I'm not sure I really see how it fits in here. It would seem, relativity and movement aside, that any distance with in the universe is a fraction of the entire size of the universe, and if the universe is infinite, then any fraction of it is infinite. Perhaps you could clarify what you mean.

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Response to Halcyon&on:
Yeah, I'm not arguing that a finite number cannot be divided infinitely, it certainly can. I'm just saying that I don't think space mirrors this.

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Revised statements:
I'm gonna leave the previous argument up there as it makes more sense considering the name of the node. After discussing the issue I see that the proof is not there. I still think it's kinda interesting though. So I now say that the universe is composed of an infinite (or near infinite) number of fundamental units of mass (I still believe in those little fellas).

Well, that's an interesting proposition, but there are some problems with it. First of all, Zeno's paradox was resolved some time ago with the mathematical concept of limits, taking an infinite sum and getting a finite number. This is an important part of Calculus. So, Zeno's paradox can't be used as an argument for a finite universe, but you really have two arguments there, not one. Your second argument- that we couldn't move if the universe was infinite -doesn't quite work because movement is relative. Though we may not be moving relative to the universe, we can still be moving relative to the Sun or the Earth or whatever you would like to move relative to. Of course, I'm only thinking philosophically here, I know nothing about the reasons in physics why the universe is or isn't infinite.

Another solution that could be applied to Zeno's paradox, at least on the macroscopic scale, is that of a discrete universe. Although calculus does say that a sum of an infinite number of components can give a finite result through integration, the science of calculus, and the concept of limits are really only theoretical constructions used to explore physical concepts.

The alternate explanation, (suggested to me by my brother), is that the space is discrete at some level. If to move x, you have to move half x, and then half that half x ... ad infinitum, instead of dividing x to inifinity, you eventually hit a wall. This is the discrete space step which you can then traverse.

Discrete space isn't any more right than calculus, but its an interesting solution. Of course, the moment you take into account quantum mechanics, the need for a discrete universe to explain Zeno's paradox becomes meaningless.

Also ... in regards to the debate above, I think that a finite number can be divided into an infinite number of parts. That is what calculus is about.

Infinity is a very tricky subject, and trying to talk about it using only our intuition is a pretty certain way to create seeming paradox, because infinity and human intuition don't match up very well.

Even if the universe is infinite, movement relative to a given point is still possible. For example, there are an infinite number of real numbers, but we still know the exact distance between any two of them.

I think the fundamental problem with this "paradox" is that it divides space infinitely, but not time. As we keep dividing the distance to travel, we must also divide the time that will be required to travel it. As the size of the distance to travel approaches 0, so to will the time required to travel it.

While Zeno's Paradox is an interesting idea that points out that you could certainly limit yourself from ever getting from any given point A to any other given point B, it's obvious that you can move from point A to point B by not choosing to incessantly continue going only half of the remaining distance between yourself and your destination, but to, instead, traverse the entire distance all at one go.

While I'm no math whiz by any stretch of the imagination, I am relatively fair with intuitive and deductive reasoning. So the two ideas that I'm about to put forth will use no math at all, but really just a small series of very simple ideas.

First I'll address the idea of motion vs. non-motion to be able to establish my second idea that the universe must be infinite in volume.

Motion of an object is generally thought to need a frame of reference. For argument's sake some refer to the object in question as the center of the universe, and that to be moving, it must change its position with regard to some other object or observable something, assuming that the other something isn't also moving relative to the object in question, or moving in an identical trajectory with the object in question with regard to all other observable things.

Let's suppose for a second that the object in question - we'll use yourself as an easy example - is the only thing in the entire universe. I assert that you may indeed move about freely whether or not you have any frame of reference. Just because there is no frame of reference does not defeat the physical reality that you indeed can move given the desire and capability to do so (assuming that you have some mysterious means of propulsion, such as being able to "swim" through a voided vacuum). The fact that there's no frame of reference by which you can observe your re-positioning, and thereby prove to yourself that you are indeed moving from one arbitrary point to another, only renders your moving around meaningless and likely pointless, but does not nullify the fact that you have indeed moved from one location to another with regard to where you were. If there was something by which you could have observed your movement, then you could be certain that you moved, but the idea that you can't be sure that you moved doesn't necessitate a fact that you didn't move. So I think you can certainly move whether or not there's any meaning or ultimate purpose to that motion.

I also assert, from a purely Newtonian mindset, that the universe must be infinite. First, I'll give the Merriam-Webster definition of universe:

universe: noun - the whole body of things and phenomena observed or postulated

And second, here's a setup to illustrate the assertion:

Let's say the universe consists of nothing but yourself, your clothing, a pebble in one pocket, a swiss army knife in your other pocket, your desire and capability to move with respect to a given center of the universe, and a handful of galaxies contained within a certain set of bounds. I'll use an over-sized shoe box as the set of bounds that contain all the matter and energy of the universe as it currently exists. And we'll say for now that the size of the box is exactly 500-zillion-bajillion light years long, along its longest side. Outside the box is nothing whatsoever - just absolute and pure void. We will further assume for now the universe has reached a complete equilibrium and is neither expanding nor contracting.

Now let's say you "swim" to the edge of the universe, cut a cantaloupe-sized hole in the box with your trusty knife, and use your thumb and forefinger to flick the pebble roughly 12 feet beyond the bounds of the universe. I realize that in the void there would be no friction to slow or stop the inertia of the pebble, but it's easier for now to just say it stopped at some arbitrarily short distance or blame some gravitational phenomenon. Now, the universe is roughly 500-zillion-bajillion light years and 12 feet long, along its longest side. You've just expanded the universe by 12 feet! Mom would be proud. The point is this: however you define the finite nature of the universe, anything outside those bounds would be non-universe, and therefore, seemingly obviously nothingness by the way we define the universe as everything that exists. And, in this case, nothingness = voided vacuum = something. Even if what's outside the universe is completely empty space, it's more or less a potential container into which matter or energy could be displaced from its current location within the shoe box universe into some location outside the shoe box, thereby changing the size, shape, consistency, etc. of the universe in its current form. The fact seems to be that even if there's nothing outside the shoe box, all of that nothingness is a part of the universe. And if there's nothing out there in the way, then you could cut a larger hole in the box, swim right up to the pebble, and kick the crap out of it to further expand the universe. I see no logical reason why you couldn't continue this action ad infinitum. The void is logically limitless and an additional part of whatever mass and energy constitutes the "meaningful" part of the universe. Thus, the universe is boundless, or infinite in volume.

As others point out, Zeno's paradox shows that given certain common-sense implicit assumptions, movement seems impossible. Quite obviously too, movement is possible; therefore one or more of the assumptions must be false.

Others have pointed out that it may be possible to pass through an infinite number of in-between locations in a finite duration of time. We can calculate the maths of that with integration and limits, but even if we couldn't, that wouldn't rule it out: not being able to prove that it is possible is quite different from proving that it is impossible.

There has been discussion on the possibility that there are not an infinite number of in-between locations. But this is known to be the case. Distance is discrete at very small scales. So is time.

The minimum distance that one can move in any meaningful sense is called the Planck length. It is around 1.6 × 10⁻³⁵ metres. This has been known for around a century - Max Planck was awarded the Nobel Prize in Physics in 1918 "In recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta." So this topic of Zeno's Paradox really is quite dead.

This says nothing about how big the universe is, or if it is infinite or not; or which kind of infinity it might be, just that it does not have infinitely small subdivisions. If the universe is a simulation, that's how big the pixels are.

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If you're not into the metric measurements for the Planck length, then 6/10000000000000000000000000000000000 of an inch is about right; if you need scientific accuracy then use the metric measurement already; that's what it's for. Put it this way: the Planck length is about 10^-20 of the diameter of a proton, which itself is about 1.65* 10⁻¹⁵ m. Shrink down to the size of a sub-atomic particle and you're not even halfway finished shrinking yet. The Planck length can be precisely described (as "1.6 × 10^-35 metres") but it really doesn't fit into the human imagination.