The 2018 Lindström Lectures was delivered by Michael Rathjen, Professor of Pure Mathematics at University of Leeds. He obtained his Ph.D. (1988) and Habilitation (1992) at University of Münster. Rathjen is renown for his fundamental contributions to Proof Theory, especially cut elimination for infinitary proof systems, ordinal analysis of impredicative theories and calibration of set-existence strength of combinatorial principles. He has also carried out penetrating investigations in intuitionistic and constructive mathematics, including Martin-Löf type theory.
The fact that “natural” theories, i.e. theories which have something like an ‘idea’ to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of mathematics. Using paradoxical methods, e.g. self-reference Rosser-style, one can distill theories with incomparable logical strengths and show that the degree structure of logical strengths is dense in that between two theories S < T one can always find a third Q such that S < Q < T. But are there ‘natural’ examples of such phenomena? We also know how to produce a stronger theory by adding the consistency of the theory. Can we get a stronger theory by adding something weaker than consistency that is still “natural”? These and other questions will be broached in the talk.
The graph minor theorem, GM, is arguably the most important theorem of graph theory. The strength of GM exceeds that of the standard classification systems of RM known as the “big five”. The plan is to survey the current knowledge about the strength of GM and other Kruskal-like principles, presenting lower and upper bounds.