Structural Complexity Measures for Foundational Theories

Formalizing observations and behavior is an integral part of science that allows us to use scientific methods to study natural phenomena. In mathematics, this is done by formulating axioms in suitable logics, such as first-order logic, which should capture the behavior of a structure or operation one wants to study. However, early results in mathematical logic tell us that this endeavor is bound to fail. It follows from Gödel`s completeness theorem that all but the most trivial axiomatizations, or theories, will have models besides the intended model whose behavior we set out to capture. Consider, for example, Peano arithmetic, the theory supposed to capture mathematics on the natural numbers; It has models with infinite numbers. How complicated are these models? Can we easily distinguish them from our intended model? Moreover, how complicated is it to describe the elements of the model which exhibit non-intended properties? This project will bring together leading experts in Scott analysis and the foundations of mathematics to answer these questions formally for so-called foundational theoriesi.e., theories in which a large part of modern mathematics can be developed.