[ozb]

Note that in general, a physical instantiation of an undecidable problem must be specified/realized to _infinite_ precision; that is, for any such system S, and for any eps>0, there is a perturbation p with distance d<eps (eg, move a billiard ball an arbitrarily small amount) that is provable; this is analogous to the fact that existence of solutions to Diophantine equations is undecidable, but the theory of real closed fields is decidable, which means that the only undecidable case is when an equation has solutions _arbitrarily close_ to integers, but never quite an integer. I am not a physicist, but I don't believe any physics actually cares about infinitely-precise setups.

[mxkopy]

Integers exist in quantum physics (e.g. electron charge, spin), which is why I think quantum gravity is important to this argument. Spacetime ends up being discretizable and we can end up having rational valued physical phenomena.

[ozb]

> integers exist

Mostly as an abstraction on top of a continuous wavefunction/quantum field

> Spacetime ends up being discretizable

As far as I know this is speculative and usually assumed by physicists to be false; it's definitely not a required feature of quantum mechanics per se, and as far as I know not of any other well-accepted theory.

[mxkopy]

> Integers are fundamental in quantum mechanics, particularly as quantum numbers that define the discrete properties of particles, such as energy levels, angular momentum, and spin.

> Quantum mechanics dictates that certain properties, like energy and angular momentum, are quantized, meaning they can only exist in discrete packets or "quanta".

This was from a cursory google search.