Gödel’s incompleteness theorems are connected to unsolvable calculations in quantum physics, and also to Turing

Davide Castelvecchi in Nature:

A logical paradox at the heart of mathematics and computer science turns out to have implications for the real world, making a basic question about matter fundamentally unanswerable.

In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its electrons — from an idealized model of its atoms.

The result also raises the possibility that a related problem in particle physics — which has a US$1-million prize attached to it — could be similarly unsolvable, says Toby Cubitt, a quantum-information theorist at University College London and one of the authors of the study.

The finding, published on 9 December in Nature¹, and in a longer, 140-page version on the arXiv preprint server², is “genuinely shocking, and probably a big surprise for almost everybody working on condensed-matter theory”, says Christian Gogolin, a quantum information theorist at the Institute of Photonic Sciences in Barcelona, Spain.

More here.