Sam Sanders - The Big, Bigger, and Biggest Five of Reverse Mathematics Part I: real analysis

This lecture was part of the Workshop on "Reverse Mathematics: New Paradigms" held at the ESI August 4 - 8, 2025. I present some recent joint work with Dag Normann (University of Oslo) on the Reverse Mathematics of real analysis and related areas. In particular, we have established the following, working in Kohlenbach's higher-order framework. (a) There are many equivalences between the second-order Big Five and basic third-order theorems from real analysis pertaining to (slightly) discontinuous functions. (b) Slight variations and generalisations of the theorems from item (a) cannot be proved from the Big Five and stronger systems. (c) The theorems from item (b) can generally be classified as equivalent to one of four `new' Big systems, namely the uncountability of the reals, the Jordan decomposition theorem, the Baire category theorem, and Tao's pigeon hole principle for the Lebesgue measure. (d) Current research includes the reverse mathematics of stronger principles (than those in (c)), namely Feferman's projection principle and the coding principle expressing that open sets of reals have second-order codes.