Projects
Project leader: Anton Freund
Title: „Continuous Order Transformations: A Bridge Between Ordinal Analysis, Reverse Mathematics, and Combinatorics“
Information on GEPRIS (DFG): https://gepris.dfg.de/gepris/projekt/460597863
Project description:
The following question is central for several branches of mathematical logic: Which axiom systems are strong enough to prove a given mathematical theorem? In addition to its intrinsic intellectual interest, an answer to this question does often yield further information about the theorem in question, for example on the quality of approximations or the complexity of algorithmic solutions.
Our project aims to deepen connections between two branches of mathematical logic, which are both concerned with the central question formulated above: ordinal analysis and reverse mathematics. As a bridge between the two approaches, we use continuous transformations (finite-type functionals) over the categories of partial and linear orders. This will allow us to answer the central question in cases where it is currently open. Specifically, we want to analyze theorems of combinatorics, mostly related to Kruskal's tree theorem, the graph minor theorem, and the theory of better quasi orders. We also aim at a general framework, in which known and new results can be explained in a uniform way.
Project leader: Anton Freund
Title: Natural Examples of Slow Independence Results in Mathematical Logic
Information on GEPRIS (DFG): https://gepris.dfg.de/project/587206874?lang=en
Project description:
In this project, we intend to find mathematically natural examples for Gödel’s incompleteness theorems that are weaker (relative to certain axiom systems) than the examples in the current literature. With reference to the slow consistency principle of Sy Friedman, Rathjen and Weiermann (2013), such examples will be called slow independence results. More specifically, it seems that no natural examples of independence are known below the Pi20-reflection principle of axiom systems. The belief that no such examples exist can be seen as a proof-theoretic variant of the famous Martin conjecture from computability theory. If successful, our project will refute this variant of the conjecture. Specifically, we aim to find a version of the Paris-Harrington principle that is strictly weaker than the classical principle but still unprovable in Peano arithmetic; to determine the proof lengths that are associated with instances of the graph minor theorem; and to study how phase transitions (in the sense of Weiermann) and sub-recursive degrees interact in the context of natural independence results.
