# Mathematical Logic

The group in Mathematical Logic specializes in the following topics: Model theory, Non-standard 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. Berarducci-Mamino showed that $\mathbf{No}$ can find applications to the asymptotic analysis of iterated exponential polynomials (Skolem functions).
A long-term project is to study Tarski’s question of whether the theory of $\mathbb{R}_{\mathsf{exp}}$ is decidable. Macintyre-Wilkie (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 o-minimal EXP-field 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 model-theoretic 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 stability-theoretic properties of their class.

Definable groups in o-minimal 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 o-minimal context and beyond, other stability-theoretic 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 o-minimal ones, theories of henselian valued fields, and more.

###### Members
Alessandro Berarducci
alessandro.berarducci@unipi.it
Marcello Mamino
marcello.mamino@unipi.it
Rosario Mennuni
rosario.mennuni@dm.unipi.it
###### Collaborators
Annalisa Conversano
Martin Hils
Salma Kuhlmann
Vincenzo Mantova
Mickaël Matusinski
##### Non-standard 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 additive-multiplicative 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.

###### Collaborators
Isaac Goldbring
Renling Jin
Lorenzo Luperi Baglini
##### Set theory and foundations

Our foundational research is devoted to the consistency of a theory of numerosity in the general context of a set-theoretic 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
Mauro Di Nasso
mauro.di.nasso@unipi.it
Marco Forti
forti@dma.unipi.it
Andreas R. Blass
##### Arithmetic and complexity

Provability logic is a modal logic tailored for the study of Gödel’s incompleteness phenomena. We propose a novel model-theoretic approach to the provability logic of Peano Arithmetic. We also deal with constraint satisfaction problems in the area of computational complexity.

###### Members
Alessandro Berarducci
alessandro.berarducci@unipi.it
Marcello Mamino
marcello.mamino@unipi.it
Manuel Bodirsky
Caterina Viola

#### People

##### Faculty
Name Surname Email Personal Card
Alessandro Berarducci
Mauro Di Nasso
Marcello Mamino
##### Affiliate Members
Name Surname Email Personal Card
Marco Forti
##### Postdoctoral Fellows
Name Surname Email Personal Card
Rosario Mennuni
##### Ph.D. Students at the University of Pisa
Name Surname Email Personal Card
Moreno Pierobon
Mariaclara Ragosta

there is no data

#### Ph.D. Theses supervised by members of the group

##### awarded by the University of Pisa
Year Name Surname Title of the Thesis Supervisor(s)
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 p-subgroups of finite groups of Lie type Marco Forti and Bertram Huppert

there is no data

#### Grants

• ##### 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
First Name Last Name Affiliation
Claudio Agostini Technische Universität Wien
Gianluca Paolini Università di Torino

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
Panteleimon Eleftheriou University of Leeds Feb 24, 2023 Mar 06, 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 degli Studi di Bologna Dec 01, 2022 Dec 03, 2022