Reverse mathematics is a fundational program whose goal is to find op- timal axioms to prove ordinary theorems. They use the framework of second- order arithmetics, with a base theory, RCA 0 , capturing computable mathematics. Started in the 70s by Harvey Friedman, they revealed five big systems of axioms such that most of the theorems coming from the heart of mathematics are either provable within RCA 0 , or provably equivalent to one of these systems.

In this talk, we will introduce reverse mathematics with its motivations, and will give a glimpse of this computational structural phenomenon. Last, we will present the new problematics of modern reverse mathematics together with its main remaining open questions.

References

1. Hirschfeldt, D.R.: Slicing the truth, Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, vol. 28. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2015), on the computable and reverse mathematics of combinatorial principles, Edited and with a foreword by Chitat Chong, Qi Feng, Theodore A. Slaman, W. Hugh Woodin and Yue Yang

2. Simpson, S.G.: Subsystems of Second Order Arithmetic. Cambridge University Press (2009)