> encode arbitrary (but not infinite) information

What leads you to claim that you can't encode infinite information given an infinite set to work with?

I suspect a looseness with language; this often happens with maths done in natural language, and I may be guilty of it even when I try not to be.

If spacetime is infinite in extent yet discrete at the Plank scale, it would be ℵ₀; the Hilbert space is real-valued and therefore (a vector of) ℵ₁.

The Bekenstein bound, in practice, would do limit how much information can be actually encoded, even if all quantum states have an underlying real value and therefore even just one quantum value of one particle is ℵ₁ all by itself.

reality does not have an infinite set though.

Infinity is an abstraction that simplifies a lot of maths, it isn't real though.