Mathematical Sciences: Proof-Theoretical Investigations of Theories

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Rathjen intends to work on the following four problems and related issues. (1) Give an ordinal analysis of Pi-1-2 comprehension. (2) Develop strong ordinal representation systems on the basis of recursively large ordinals in lieu of their large cardinal analogues. (3) Determine the proof-theoretic strength of restricted foundations in weak set theories. (4) Is Feferman's theory T-0 together with least fixed point axioms for monotone operators conservative over T-0 with respect to arithmetic sentences? Proof theory concerns itself with axiomatic systems and methods of proof within them, particularly with the relative strength of different systems. The system A is said to be properly stronger than the system B if every proposition that has a proof in B also has a proof in A but the converse is not so. The investigator is pursuing a number of questions along these lines, varying in technical difficulty and degree of interest to the experts, but none is really susceptible to a brief discussion in laymen's terms.