Interesting discussion from *NOVA:*

Zeno’s paradox is solved, but the question of whether there is a smallest unit of length hasn’t gone away. Today, some physicists think that the existence of an absolute minimum length could help avoid another kind of logical nonsense; the infinities that arise when physicists make attempts at a quantum version of Einstein’s General Relativity, that is, a theory of “quantum gravity.” When physicists attempted to calculate probabilities in the new theory, the integrals just returned infinity, a result that couldn’t be more useless. In this case, the infinities were not mistakes but demonstrably a consequence of applying the rules of quantum theory to gravity. But by positing a smallest unit of length, just like Zeno did, theorists can reduce the infinities to manageable finite numbers. And one way to get a finite length is to chop up space and time into chunks, thereby making it discrete: Zeno would be pleased.

He would also be confused. While almost all approaches to quantum gravity bring in a minimal length one way or the other, not all approaches do so by means of “discretization”—that is, by “chunking” space and time. In some theories of quantum gravity, the minimal length emerges from a “resolution limit,” without the need of discreteness. Think of studying samples with a microscope, for example. Magnify too much, and you encounter a resolution-limit beyond which images remain blurry. And if you zoom into a digital photo, you eventually see single pixels: further zooming will not reveal any more detail. In both cases there is a limit to resolution, but only in the latter case is it due to discretization. More.

This is a self-contradictory statement. But NOVA has always been partial to Star Trek voodoo physics. You know, time travel, warp speed and all that nonsense.

“Is there a smallest unit of length?”

Yes

Finally, after all these years I get to say it!

The smallest unit of length is the distance one must move the hot tap in the shower while attempting to subtly increase the temperature of the water to acceptable warmth before one’s skin begins to peel off in a blistering supernova.

I am content.

I also have a definition for the smallest unit of time but it must wait for another – time.

I’ve always been biased to the notion of a smallest unit of length. Infinite infinitesimals don’t seem to add up… funny… an unintended pun. 😉

Infinity is not a logical concept for reasons that I don’t have the time to into right now. In fact, Zeno’s paradoxes are really simple but powerful refutations of all that infinity and continuity nonsense.

It follows that the universe is discrete, i.e., there is a fundamental unit of length, energy and motion. Max Planck pretty much proved this empirically with his quantum energy experiments and calculations. He’s the father of quantum physics, IMO. The only reason that continuity is still a part of physics is that it is also an inherent part of Einsteinian physics. You can’t have black holes without continuity. And the physics community is proud of their black holes. Besides, Einsteinian physics is a mighty fortress indeed. Only a Kuhnian revolution can put it to rest for good. That will come soon enough.

The “Planck Length” is the smallest unit known to man. It’s really dinky.

https://en.m.wikipedia.org/wiki/Planck_length

“The size of the Planck length can be visualized as follows: if a particle or dot about 0.1 mm in size (which is approximately the smallest the unaided human eye can see) were magnified in size to be as large as the observable universe, then inside that universe-sized “dot”, the Planck length would be roughly the size of an actual 0.1 mm dot. In other words, a 0.1 mm dot is halfway between the Planck length and the size of the observable universe on a logarithmic scale.”