Aspects of undecidability

Juan Pablo Aguilera is a researcher at the Vienna University of Technology in Austria. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Philosophy, Linguistics and Theory of Science, University of Gothenburg.

According to Gödel’s incompleteness theorem, mathematics is incomplete. This means that there are mathematical questions that can be asked, but not answered. Mathematical logic attempts to understand this phenomenon through a variety of approaches. Proof theory is the study of mathematical axioms, proofs, and theories. Set theory is the study of infinity and the limits of mathematics. Computability theory is the study of algorithms and impossibility. These three fields together provide a broad perspective of undecidability.

This project will study the problem of undecidability and its connections to mathematics, computer science, and philosophy. The role of philosophy is to clarify the impact of undecidability in mathematics, to what degree it is unavoidable, and its significance for our conception of science. In computer science, we try to produce frameworks and algorithms that avoid this phenomenon in order to ensure its reliability.

Finally, with mathematics we attempt to quantify the degree of unsolvability of problems and whether it can lead to new kinds of mathematics, or shed new lights into old results.

Photo: Ulrich Zinell