Proofs Beyond the Transfinite

In 1931, Austrian mathematician Kurt Gödel first showed that some mathematical problems are formally unsolvable. This means that there exist mathematical questions for which mathematics cannot provide any answers. Proof Theory is the branch of mathematical logic which studies unsolvable problems, mathematical theories, and the existence of proofs. Since Gödels work, the field has evolved rapidly. Ordinal Analysis uses methods related to infinitary proofs and transfinite numbers to study whether mathematical problems have answers, classify theories, and understand the structure of possible mathematical proofs. However, its scope is limited to a certain particular kind of mathematical problems. The goal of this project is to extend Ordinal Analysis to more general classes of problems in order to develop new tools to study mathematical theories and determine which mathematical questions have answers and which do not.