The group in Mathematical Logic specializes in the following topics: Model theory, Nonstandard methods in Ramsey theory and combinatorics, Set theory and Foundations, Arithmetic, and complexity. Details on these topics can be found below.
The group runs a Seminar Series: further information can be found at this link.
Research Topics
Model theory
We study the model theory of ordered fields endowed with an exponential function and/or a derivation and/or a valuation. Typical examples are the real exponential field $\mathbb{R}_{\mathsf{exp}}$, the differential field $\mathbb T$ of transseries, Conway’s field $\mathbf{No}$ of surreal numbers, Hahn’s fields of generalized series. These subjects are strictly related. $\mathbb{R}_{\mathsf{exp}}$ is elementarily equivalent to $(\mathbf{No},\mathsf{exp})$ where $\mathsf{exp}\colon \mathbf{No} \to \mathbf{No}$ is the exponential map of Kruskal and Gonshor. Moreover $\mathbb T$ is is elementarily equivalent to $(\mathbf{No},\partial)$ where $\partial\colon \mathbf{No} \to \mathbf{No}$ is the derivation introduced by Berarducci and Mantova. BerarducciMamino showed that $\mathbf{No}$ can find applications to the asymptotic analysis of iterated exponential polynomials (Skolem functions).
A longterm project is to study Tarski’s question of whether the theory of $\mathbb{R}_{\mathsf{exp}}$ is decidable. MacintyreWilkie (1996) proved the decidability assuming Schanuel’s conjecture (SC) in transcendental number theory.
Building on this and the work of various collaborators, we intend to address the problem of whether, assuming SC, any ominimal EXPfield is elementary equivalent to $\mathbb{R}_{\mathsf{exp}}$. We expect that the study of the residue structure (with respect to the natural valuation) can provide important information.
We also concentrate on another line of research in the model theory of ordered structures, namely the study of ordered abelian groups equipped with an automorphism, with a special focus on the existentially closed ones.
The latter form a class which is not elementary, but to which modeltheoretic techniques may nevertheless be applied, by using positive logic in the sense of Ben Yaacov and Poizat (2007). Current work in progress deals with the amalgamation of ordered abelian groups with an automorphism, and with generalized stabilitytheoretic properties of their class.
Definable groups in ominimal structures, which present striking analogies with real Lie groups, can be studied by using generalisations of techniques which originally arose in stability theory, such as generic types and connected components. In the ominimal context and beyond, other stabilitytheoretic notions which can be used on suitable spaces of types, e.g. global invariant ones, include the Morley product, orthogonality, and domination. We investigate the compatibility of domination with the Morley product and, when compatibility holds, the structure of the resulting quotient semigroup, the domination monoid, in various classes of theories, such as the ominimal ones, theories of henselian valued fields, and more.
Members
Collaborators
Nonstandard methods in Ramsey theory and combinatorics
In recent years, a new line of research has emerged in which the problems of Ramsey Theory and additive combinatorics are studied using the perspective and methods of nonstandard analysis. In particular, we focus our research in two directions. The first direction is Arithmetic Ramsey Theory, where one looks for monochromatic configurations which are found in every finite coloring of the integers. Two main open problems in this context are the existence of monochromatic Pythagorean triples $a,b,c$ such that $a^2 + b^2 = c^2$, and of monochromatic additivemultiplicative quadruples $a, b, a + b, a \cdot b$. The second direction is the area of additive combinatorics where one looks for arithmetic configurations which are found in every set of integers of positive asymptotic density.
Members
Collaborators
Set theory and foundations
Our foundational research is devoted to the consistency of a theory of numerosity in the general context of a settheoretic universe. Numerosity Theory aims to generalize cardinality in order to maintain the property that proper subsets have smaller numerosity. Starting from the construction of models for the theory of numerosity, several problems have naturally emerged about the existence of ultrafilters with special properties. Also in Arithmetic Ramsey Theory, ultrafilters with peculiar properties are often used, such as idempotent and minimal ultrafilters. Our research in Set Theory and Foundations focuses on the construction of models for the numerosity theory, and on the study of the aforementioned special classes of ultrafilters.
Members
Collaborators
Arithmetic and complexity
Provability logic is a modal logic tailored for the study of Gödel’s incompleteness phenomena. We propose a novel modeltheoretic approach to the provability logic of Peano Arithmetic. We also deal with constraint satisfaction problems in the area of computational complexity.
Members
Collaborators
People
Faculty
Name  Surname  Personal Card  

Alessandro  Berarducci  alessandro.berarducci@unipi.it  
Mauro  Di Nasso  mauro.di.nasso@unipi.it  
Marcello  Mamino  marcello.mamino@unipi.it 
Affiliate Members
Name  Surname  Personal Card  

Marco  Forti  forti@dma.unipi.it 
Former Members
there is no data
Postdoctoral Fellows
Name  Surname  Personal Card  

Rosario  Mennuni  rosario.mennuni@dm.unipi.it 
Ph.D. Students at the University of Pisa
Name  Surname  Personal Card  

Mariaclara  Ragosta  mariaclara.ragosta@phd.unipi.it 
Ph.D. Students at other institutions
Name  Surname  Affiliation 

Paolo  Arruda  Universität Wien 
Ph.D. Theses supervised by members of the group
awarded by the University of Pisa
Year  Name  Surname  Title of the Thesis  Supervisor(s) 

2023  Moreno  Pierobon  Nonstandard and boolean valued models, and interactions with combinatorics, dynamics, and sheaf theory  Mauro Di Nasso 
2019  Andrea  Vaccaro  $\mathbb{C}^\ast$algebras and the Uncountable: a systematic study of the combinatorics of the uncountable in the noncommutative framework  Alessandro Berarducci and Ilijas Farah 
1995  Andrea  Previtali  Structure and character degrees of sylow psubgroups of finite groups of Lie type  Marco Forti and Bertram Huppert 
awarded by another institution
Year  Name  Surname  Title of the Thesis  Institution  Supervisor(s) 

2012  Lorenzo  Luperi Baglini  Hyperintegers and Nonstandard Techniques in Combinatorics of Numbers  Università degli Studi di Siena  Mauro Di Nasso 
Grants
Current
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Past

Mathematical Logic: models, sets, computability (Prin 2017)
Principal Investigator: Alessandro Berarducci
Project period: Aug 01, 2019 – Aug 19, 2023

Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni (Progetti di Ricerca di Ateneo (PRA) 2020  2021)
Principal Investigator: Giacomo Tommei
Project period: Jul 07, 2020 – Dec 31, 2022

Sistemi dinamici in analisi, geometria, logica e meccanica celeste (Progetti di Ricerca di Ateneo (PRA) 2017  2018)
Principal Investigator: Marco Abate
Project period: Apr 10, 2017 – Apr 09, 2019
Visitors
Prospective
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Current
there is no data
Grouped by year
2023
First Name  Last Name  Affiliation  From  To 

Claudio  Agostini  Technische Universität Wien  Apr 27, 2023  Apr 28, 2023 
Paola  D'Aquino  Università degli Studi della Campania "Luigi Vanvitelli"  Mar 30, 2023  Mar 30, 2023 
Panteleimon  Eleftheriou  University of Leeds  Feb 24, 2023  Mar 06, 2023 
Sonia  L'Innocente  Università di Camerino  May 31, 2023  Jun 01, 2023 
Gianluca  Paolini  Università di Torino  May 11, 2023  May 13, 2023 
2022
First Name  Last Name  Affiliation  From  To 

Jan  Dobrowolski  University of Manchester  Nov 13, 2022  Nov 19, 2022 
Martino  Lupini  Università di Bologna  Dec 01, 2022  Dec 03, 2022 