To be clear the limitation here isn't silicon logic, it's theoretical logic (specifically general recursive functions). It sure seems like any possible computation can be expressed as a general recursive function, but that's a scientific thesis, not a mathematical theorem. As we have yet to formalize "define a physical system" it is possible that this task isn't actually expressible in 21st century mathematical logic. I suppose some custom hardware which doesn't use logic at all might help, but then Gödel's theorem wouldn't apply at all. (Likewise with modal logic.)