Why, how do you do?

[Tiberius]

Since [0,1] is uncountable, there exist uncountably many points along any finite length in the universe. Why is this not a counterexample to your statement in bold?

Because there is no evidence to suggest that space can be divided into an infinite amount of sections. At some point, you reach atomic lengths, then the length of electrons, then...etc. Essentially there has to be some standard "smallest" length, rather than an infinitely small one, because otherwise physics breaks down and you get paradoxes.

Zeno came up with a few of them, namely Achilles and the Tortoise (dealing with space):

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

and the arrow paradox (dealing with time):

Zeno states that for motion to be occurring, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one instant of time, for the arrow to be moving it must either move to where it is, or it must move to where it is not. However, it cannot move to where it is not, because this is a single instant, and it cannot move to where it is because it is already there. In other words, in any instant of time there is no motion occurring, because an instant is a snapshot. Therefore, if it cannot move in a single instant it cannot move in any instant, making any motion impossible.