Undecidability in physical systems

Since the works of Gödel and Turing, we know that there exist true mathematical statements that can neither be proved nor disproved. As we use mathematics to describe the physical world, it’s natural to ask whether similar limits arise in physics. This topic has gained renewed attention in recent years, driven by the developments in quantum information theory. After a short overview of undecidability, we will see that some physical models can indeed show undecidable behaviour — for instance, in quantum many-body systems, the spectral gap problem is undecidable.