Marta Fiori Carones


Ph.D Cycle
XXXII
Supervisor
Alberto Marcone
Institutional Office Address
DMIF, Room NN2, Phone: +39 0432 558469
Research Project Title
Order theory and graph theory in reverse mathematics
Research Project Description
My research project focuses on some aspects of order theory and graph theory in reverse mathematics. The first results in reverse mathematics date back to the seventies, but it is still an active field in the context of mathematical logic. Reverse mathematics developed within the formal framework of second order arithmetic, a theory powerful enough to prove many theorems of ordinary mathematics. Moreover, this formal theory allows to understand the exact amount of set existence axioms needed in order to prove a great amount of mathematical theorems. This peculiarity of reverse mathematics provides some results that open also some questions in the area of foundations of mathematics. At the same time the subtheories of second order arithmetic, to whom some theorems are equivalent, yields also a hierarchy of these theorems. On this light, reverse mathematics links with other, similar to some extent, hierarchies of mathematical theorems, e.g. computable or Weihrauch hierarchy. The interplay between these perspectives provides a subtler
analysis of the principles involved and a development of the same project of reverse mathematics.
Many statements of different fields of mathematics have already been analyzed. Some research has already been done in ordered set theory too, but in our opinion there are some yet unexplored areas. In particular, we are working on interval orders and interval graphs and on induced ordering of power sets of ordered sets.