In 1931, Kurt Gödel published his incompleteness theorems, which shattered mathematicians’ hopes of finding a solid foundation for mathematics. These theorems revealed that any set of axioms that could serve as a foundation for math will always be incomplete, and no set of axioms can prove its own consistency. Gödel’s ideas have led to the discovery of unanswerable questions in both math and physics, suggesting that his incompleteness theorems affect not just math, but reality itself. Gödel’s main technique in proving these theorems was to map statements about a system of axioms onto statements about numbers, allowing the system of axioms to talk about itself. Through this mapping, he demonstrated the limitations of mathematical knowledge and the impossibility of finding a complete mathematical theory of everything.
https://www.quantamagazine.org/how-godels-proof-works-20200714/