As far as we can tell, GR implies, and we have measured, space-time is completely continuous. Draw a square on a piece of paper; or, better yet, outline a cube with some sticks: within that square (or cube) is an infinite set of points of either the integral or real cardinality — whichever you’d like.
The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.
To my mind GR (or at least the standard textbook version of it, anyway) _assumes_ that space-time is continuous, it does not _imply_ it as such. Continuity is baked into the foundations of any physical theory that is expressed in the language of differential equations.
You probably know this, but it's easy to confuse the map (a physical theory) with the territory (reality, which is far more complicated).
No, physicists do not think that space is like a tinily subdivided grid. Lots of physics would break if that were the case, including QM.
We have a lot of hints that the amount of information in any given space might be bounded, but yet that space appears continuous. How exactly you reconcile this is (one of) the mystery of quantum gravity.
Sure, we cannot measure infinity, but to be fair, all mathematical concepts (when looked at closely enough) are not something we measure directly.
Even if a kindergarten-level maths of "there are three apples," we do an abstraction. We need to decide that something is a separate object, an apple (how big or small should a fruit be an apple? if there is a bite, is it an apple? etc, etc) - usually with an assumption that all apples are the same (which we know is not true, but serves as an useful approximation).
pretend that
> David Hilbert famously argued that infinity cannot exist in physical reality. The consequence of this statement — still under debate today — has far-reaching implications.
The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.