Reverse Mathematics and Degrees of Unsolvability

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

Reverse mathematics is a far-reaching research program in the foundations of mathematics, wherein core mathematical theorems are classified up to logical equivalence according to which set-existence axioms are needed to prove them. The degrees of unsolvability are a well known algebraic structure arising from basic notions of relative computability and algorithmic unsolvability. In the current research project, some outstanding problems in the reverse mathematics of measure theory are being addressed in unexpected ways, by applying recent advances in algorithmic randomness and Kolmogorov complexity. In the realm of general topology, reverse mathematics itself is being extended far beyond the standard so-called "big five" equivalence classes, to encompass much stronger systems. Remarkable relationships are being revealed between, on the one hand, reverse mathematics and other foundationally interesting topics, and on the other hand, degrees of unsolvability of mass problems associated to effectively closed sets of real numbers. Reverse mathematics is a far-reaching research program in the foundations of mathematics. The purpose of reverse mathematics is to elucidate the axiomatic and logical structure of mathematics as a whole. It turns out that many core mathematical theorems fall into a relatively small number of logical classes, the so-called "big five" classes. Thus one sees a remarkably orderly structure. The purpose of the current research project is to further clarify and extend this structure. Among the technical tools being deployed are algorithmic randomness, complexity theory, and degrees of unsolvability.