Physics undecidable problem in ZFC.

Question

Is there a physical problem that is undecidable in Zermelo-Fraenkel-Choice set theory? Something related with free abelian groups and Whitehead problem perhaps?

Answer 1

There are numerous questions about the nature of the solutions to specific differential equations that are computationally undecidable, and which therefore admit numerous specific instances whose solution has a nature independent of ZFC or of any other fixed consistent theory.

For example, in the paper Boundedness of the domain of definition is undecidable for polynomial ODEs, the authors Graca, Buescu and Campagnolo prove that the question of whether the differential equation $\frac{dx}{dt}=p(t,x)$ with initial conditiion $x(t_0)=x_0$, where $p$ is a vector of polynomials, has a solution with unbounded domain or not, is computationally undecidable.

My point is that whenever a problem like this is computationally undecidable, then it follows that infinitely many specific instances of it are also provably undecidable in any fixed consistent true theory, such as PA or ZFC (or ZFC + large cardinals). The reason is that if a consistent true theory were able to settle all but finitely many instances of the question, then the original problem would be decidable by the algorithm that simply searched for proofs. One can write down a very specific polynomial ODE, such that one cannot prove or refute in ZFC whether it has an unbounded solution or not.

I think there are many other similar examples. I recall hearing in my graduate student days about similar examples, such as the question of whether a given dynamical system is chaotic or not, is also undecidable in general. Therefore these other questions also admit numerous instances of ZFC independence.