March - June 2024, "Numbers, Numerosities and Non-Standard Analysis", V. Benci (University of Pisa).
Course duration: 8 h.CFU: 2.
The schedule of the course is tentative and will take into account the requests of the participants.
In this course the notions of Non-Standard Analysis are introduced starting from the notion of number and enumeration systems. This approach is certainly not the most direct, but it provides a reflection on the notion of number and more generally on the foundations of mathematics.
May - July 2024, "Introduction to the Zoo of Reverse Mathematics", A. Marcone (University of Udine).
Course duration: 12 h.CFU: 3.
The schedule of the course is tentative and will take into account the requests of the participants.
The course will explore some of the results at the basis of the explosion of the zoo of reverse mathematics. Using computable forcing to build appropriate omega-models we will prove that Weak Konig Lemma (WKL) and Ramsey Theorem for pairs (RT22) do not imply each other. We thus concentrate on Seetapun’s theorem (RT22 does not imply arithmetical comprehension) and Liu’s theorem (RT22 does not imply WKL).
May - June 2024, "Linear Orderings", T. Weinert (University of Udine).
Course duration: 16 h.CFU: 4.
The schedule of the course is tentative and will take into account the requests of the participants.
Linear Orderings are a quite general class of orderings which are mathematically interesting from more than one vantage point. They provide important examples in different fields of logic such as model theory and reverse mathematics and one of the most famous problems in the history of set theory, Souslin’s problem is concerned with linear orderings. The lecture going to cover mostly combinatorial aspects of linear orderings. These are going to include but might not be restricted to combinatorics of partitions of linear orderings, in particular their Ramsey theory. Other aspect might include problems involving sums or products of ordinals. Depending on the available time, a few theorems might be stated without proof.
January - February 2024, "Determinacy and I0", V. Dimonte (University of Udine).
Course duration: 12 h.CFU: 3.
The schedule of the course is tentative and will take into account the requests of the participants.
There are many similarities between I0, the “largest” large cardinal, and the Axiom of Determinacy. In this advanced course we will first see some consequences of the Axiom of Determinacy (like measurability, the Coding Lemma and some specific partition property), and then we will see that under I0 a generalization of such consequences holds.
May - July 2023, "Reverse Mathematics", A. Marcone (University of Udine).
Course duration: 16 h.CFU: 4.
The schedule of the course is tentative and will take into account the requests of the participants.
The course will deal with the following topics: -Computable mathematics and problem reducibility -Second order arithmetic and its subsystems
January 2022, "Infinite games and (generic) large cardinals", P. Holy (Udine).
Course duration: 12 h.CFU: 3.
The schedule of the course is tentative and will take into account the requests of the participants.
In this course, we will investigate particular infinite games — Ulam style cut and choose games and variants of the precipitous game — and how the existence of winning strategies for particular players in these games is connected to generic large cardinals, and to the existence of large cardinals. The main results of this course will show how to obtain, starting from large cardinals, winning strategies in these games on small cardinals, and on the other hand, how to obtain (generic) large cardinals from winning strategies in these games.
June - July 2021, "Determinacy and large cardinals", V. Dimonte (Udine).
Course duration: 12 h.CFU: 3.
The schedule of the course is tentative and will take into account the requests of the participants.
The objective of this PhD course is to show connections between large cardinals and the Axiom of Determinacy, and more specifically to show that large cardinals are needed for AD. As the techniques used come from computability theory, descriptive set theory, and large cardinals, this is a good example of how the three disciplines cooperate and complement each other.
January - February 2021, "Model Theory", S. Barbina (The Open University).
Course duration: 16 h.
July 2020, "Reverse Mathematics of Combinatorial Principles", D. Dzhafarov (University of Connecticut).
Course duration: 10 h.
June 2020, "Generalized Descriptive Set Theory II", V. Dimonte (Udine).
Course duration: 10 h.CFU: 2.
The schedule of the course will take into account the requests of participants.The course is a continuation of the course offered by prof. Motto Ros in Torino (30h), but it is self-contained.
Measurable cardinals. Prikry forcing and strong Prikry condition. G_0 dichotomy and Silver Dichotomy for \lambda-Polish spaces.
March - April 2020, "Model Theory", S. Barbina (The Open University).
Course duration: 16 h.
ATTENTION: Course postponed due to the Covid-19 outbreak.
March - April 2019, "Teoria descrittiva degli insiemi e forcing", V. Dimonte (Udine), A. Marcone (Udine).
Course duration: 24 h.