Ph.D XXXVIII

Supervisor: Alberto Marcone

Phone: +39 0432 558401

Room: Rizzi L2-07-ND

Mail: volpi.andrea@spes.uniud.it

Research Project

Reverse Mathematics of Posets and the Transfinite Ramsey Theorem

Reverse Mathematics is a branch of mathematical logic which deals with the equivalence between mathematical theorems and axioms. The most suitable environment for this kind of work is Second Order Arithmetic and its subsystems, since it is possible to formalize most of “ordinary” mathematics there. The goal of Reverse Mathematics is to determine which set existence axioms are sufficient and necessary to prove a mathematical theorem. The project started in the 1970s with the work of Harvey Friedman and gained popularity thanks to Stephen Simpson’s work in the 1980s. After 40 years, results of many different areas of mathematics have been studied from the perspective of Reverse Mathematics. My research mostly focuses on the reverse mathematics of results about the dimension theory of partially ordered sets.

I am also planning to study the extension of the finite Ramsey Theorem to transfinite notions of largeness. In fact, under certain circumstances, it is possible to define when a finite subset of natural numbers is large with respect to a countable ordinal. This idea started with the Paris-Harrington Theorem and a conjecture dealing with a very general situation has been formulated by Alberto Marcone and Antonio Montalbán using the Veblen functions.