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Quantum mechanics and uncertainty
The uncertainty principle is a physical limit: a particle does not have a sharp position and momentum at the same time. Gödel is a logical limit on formal systems. They are limits of different kinds that share one lesson, that a complete, fully determined description does not exist. Both broke an early-1900s dream (Laplace’s predictable universe, Hilbert’s complete mathematics).
There is one real bridge. In 2015, Cubitt, Perez-Garcia and Wolf showed that for certain quantum materials, whether the system has an energy gap is undecidable. Other links, like Penrose’s, are speculation that most logicians reject.