Vittorio Cipriani


Supervisor: Alberto Marcone

+39 0432 558401

Stanza / Room: L2-07-ND (DMIF, NS2)

Research Project

Reverse Mathematics and Weihrauch Hierarchy

My research is focused on the connection between Reverse Mathematics and Weihrauch hierarchy. Reverse Mathematics studies which axioms are required to prove theorems of mathematics. These theorems are formalized in the language of second-order arithemtic, a theory strong enough to prove many theorems of ordinary mathematics. To prove that a system of axioms S is necessary to prove a theorem T, we first show that T is provable from S and then we show that T implies S, using only tools available in the specific S. Many theorems turned out to be equivalent to one of the so called “big five”: $RCA_0$, $WKL_0$, $ACA_0$, $ATR_0$ and $\Pi_1^1-CA_0$. Reverse mathematics has many similarities with other hierachies like the Weihrauch hierarchy. In this context, Weihrauch reducibility is a tool that helps in comparing the computational content of mathematical theorems, that otherwise would have been at the same level from the viewpoint of reverse mathematics. In other words, it helps us to have a “finer” classification. Anyway, so far, Weihrauch reducibility has been mainly used at the lower levels of reverse mathematics and it would be interesting to study it in the higher levels.