Say for example someone shoots an arrow at a target. At some point, the distance between the shooter and the target must be half of what it originally was. If that distance keeps being cut in half, how can two objects ever actually touch? I'm sure there's a theory for this in math or something but its mind blowing to think about

It's called Zeno's paradox and there have been many proposed solutions, but I think the best is the one that notes that the problem itself sort of evaporates when put into the context of quantum mechanics. "It's been noted that Zeno's "Arrow" argument could also be made in the context of continuous motion, where in any single slice of time there is (presumed to be) no physical difference between a moving and a non-moving arrow. Thus, Zeno suggests that if all time is composed of instants (continuous or discrete), and motion cannot exist in any instant, then motion cannot exist at all. A naive response to this argument is to point out that although the value of a function f(t) is constant for a given t, the function f(t) may be non-constant at t. But, again, this explanation doesn't really address the phenomenological issue raised by Zeno's argument. A continuous function (as emphasized by Weierstrass) is a static completed entity, so by invoking this model we are essentially agreeing with Parmenides that physical motion does not truly exist, and is just an illusion, i.e., "opinions", arising from our psychological experience of a static unchanging reality. Of course, to accomplish this we have expanded our concept of "the existing world" to include another dimension. If, instead, we insist on adhering to the view of the entire physical world as a purely spatial expanse, existing in and progressing through a sequence of instants, then we again run into the problem of how a quality that exists only over a range of instants can be causally conveyed through any given instant in which it has no form of existence. Before blithely dismissing this concern as non-sensical, it's worth noting that modern physics has concluded (along with Zeno) that the classical image of space and time was fundamentally wrong, and in fact motion would not be possible in a universe constructed according to the classical model. We now recognize that position and momentum are incompatible variables, in the sense that an exact determination of either one of them causes the other to be completely indeterminate. According to quantum mechanics, the eigenvalues of spatial position are incompatible with the eigenvalues of momentum so, just as Zeno's arguments suggest, it really is inconceivable for an object to exhibit a definite position and momentum (motion) simultaneously. The theory of special relativity answers Zeno's concern over the lack of an instantaneous difference between a moving and a non-moving arrow by positing a fundamental re-structuring the basic way in which space and time fit together, such that there really is an instantaneous difference between a moving and a non-moving object, insofar as it makes sense to speak of "an instant" of a physical system with mutually moving elements. Objects in relative motion have different planes of simultaneity, with all the familiar relativistic consequences, so not only does a moving object look different to the world, but the world looks different to a moving object. This resolution of the paradox of motion presumably never occurred to Zeno, but it's no exaggeration to say that special relativity vindicates Zeno's skepticism and physical intuition about the nature of motion. He was correct that instantaneous velocity in the context of absolute space and absolute time does not correspond to physical reality, and probably doesn't even make sense." -www.mathpages.com/rr/s3-07/3-07.htm Other proposed solutions here: http://en.wikipedia.org/wiki/Zeno%27s_paradox#Proposed_solutions

Zeno's paradox depends on the concept of absoloute and infinite space for it to work... and since space is not infinite, the paradox is illegitimate. .

Infinitely divisible space, yes...but that is not the same thing as space that is infinite in extension, or distance. We can take an inch and make infinite divisions along it, but we wouldn't say that an inch is infinitely long...

The object isn't being propelled just half way in between the target, you're thinking too deep. I do understand what you're saying, then again you're making no sense.

How is space that is infinite in extension not the same as infinitely divisible space? If it extends infinitely, or is extended infinitely, can it not be divided infinitely?

I showed you why they are not the same. An inch is a finite space, but we can still divide it in an infinite number of ways, if we are assuming that there is no fundamental quantization of space-time (in other words, there is no length of time, or space, that is fundamentally the shortest possible length of time.) Therefore, under such a conceptualization of the universe (which is rapidly falling out of step with modern physics,) an infinite space can be divided infinitely, but so can a finite one. Think of it kind of like a camera zooming out to infinity, versus zooming infinitely in. We can take a known, finite length like a foot or an inch and we can zoom in infinitely (according to Zeno,) but we can't zoom out. We see that it's a foot or an inch at some point and any further than that, it stops being a foot or an inch. With measurements like those it seems sorta like, "well fuck it let's just zoom out more and stop calling it an inch," but Zeno was talking about the distance between an arrow and its target. There's a real limit on that for real, physical reasons that we can experience. Going in the other direction, not so much (at least in Zeno's time.)

Regardless of Zeno's Paradox, two objects can never touch because of the magnetic forces that repel electrons. One atom can never touch another atom because the two atoms would then have to go through each other's electron clouds, which is not possible because the electron clouds repel each other. Imagine that you have two magnets that repel each other. It is the same concept, but atom sized and multiplied by trillions.

Not trying to sound like an ass, but how the fuck can anyone make the claim that space is limited or unlimited. Come to think of it, what is space and how the fuck does existence supposedly exist?

technically, nothing ever touches anything...if you loook at it on a small enough scale edit: Smashx420 said it

Actually no two objects ever directly touch. In fact you aren't even touching the chair you are sitting on. At an atomic level your particles and the particles of the surface you are "touching" actually repel each other. So in essence you are technically floating above the chair you are sitting on. I can't recall what this is called. But it was also stated above.

Penetration can still happen with magnetic repulsion, there just needs to be enough force in the correct direction.