Comment by jfengel

We cannot measure anywhere near the Planck length or anything like 100 significant figures.

My concern with the reals goes the other direction. The reals are not closed under common operations. A lot of the work is done in complex numbers, which are. The rationals are not closed under radicals, and radicals seem pretty fundamental.

You can define a finitist model despite this. But it's ugly, and while ugliness is not physically meaningful, it tends to make progress difficult. A usable solution may yet arise; we shall (perhaps) see.