Is there a shortest possible distance besides zero? Or there's no limitation at all?
Good question. Check out my discussion of it here.
Strictly mathematically, both of your alternatives appear to be incorrect.
There is always a smaller nonzero distance -- just keep dividing by two.
There is also a limit: zero.
But is there a smallest meaningful nonzero distance? That is, some distance that no possible event can occur over a smaller distance, no matter how little time it takes?
Sure. That's called a Planck Length. Planck units are the "smallest meaningful" amounts of anything. Planck time is the smallest meaningful amount of time -- nothing can occur in less time, for example.
The Planck Length is something like 0.0000000000000000000000000000000000016 meters.
That's how far a photon traveling at c will travel in one unit of Planck Time. A proton is about 100,000,000,000,000,000,000 Planck Length units across.
Is there a distance shorter than one Planck Length? Can't you just divide that by two, and so on? Not meaningfully, no. You can conceive of a smaller length philosophically in your head, but that cannot express a valid measurement in the real world.
What will happen when a matter travel for a distance equal to Planck Length? The matter can't travel for any distance between Plack Length and zero, can it?
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