but

> Popescu objects to the idea that digits of real numbers count as information.

I don't know where to stand, what about information encoded in geometry, like pi. If I get a spherical system, in a small enough space - no, in any space - then there's a "cutoff" to the actual number of digits to pi. Because a chunk of spacetime can't contain infinite information? sounds good.

> Quantum math bundles energy and other quantities into packets, which are more like whole numbers rather than a continuum. And infinite numbers get truncated inside black holes.

Layman here, but AFAIK concepts like black holes aren't consistent with quantum mechanics so I'm not sure it's wise to use concepts from both theories at the same time. (i.e QM predicts that wave functions evolve deterministically but GR predicts information loss in black holes, these two views conflict).

It's way beyond my grasp but some theories seems to quantize space, I wonder how those agree with the notion of "thickness".

I'm disappointed that the article doesn't point into any mathematical theory that models the "thickness" that comes from removing the empty middle theorem.