Reverse Mathematics beyond the Gödel hierarchy

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David Hilbert’s famous list of 23 open problems, presented at the Paris ICM in 1900, contains a number of foundational problems: e.g. Problem 2 pertains to the consistency of mathematics, i.e. the fact that no contradiction can be proved. Hilbert later developed Problem 2 into Hilbert’s program for the foundations of mathematics, but Gödel’s famous incompleteness theorems show that this program is impossible. As a positive outgrowth, Hilbert’s notion of consistency gave rise to the Gödel hierarchy, a linear order that is said to capture essentially all natural and significant logical systems. Nonetheless, together with Dag Normann, I have recently identified a significant number of basic and natural theorems of uncountable mathematics (like the Heine-Borel compactness of the unit interval) that fall outside of the Gödel hierarchy. The aim of this project is to obtain a large collection of theorems outside of the Gödel hierarchy that form a parallel hierarchy. To this end, I will develop the following topics in Reverse Mathematics, which is a foundational program that seeks to identify the minimal axioms needed to prove theorems of ordinary mathematics.(T.1) The Reverse Mathematics of measure and integration theory, with a focus on the gauge integral. (T.2) The Reverse Mathematics of topology, with a focus on robust results.(T.3) New classes of theorems in Reverse Mathematics: uniformity, splittings, and disjunctions.Topics (T.1)-(T.3) will provide a large collection of highly natural theorems of mathematics outside the Gödel hierarchy. These topics split into sub-topics that naturally connect to and extend existing research in Reverse Mathematics. Finally, Kohlenbach's Higher-order Reverse Mathematics provides the most natural framework for the study of (T.1)-(T.3).