# Mathematical Logic

The group of 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
• Marcello Mamino
• Rosario Mennuni
##### Collaborators:
• Mickaël Matusinski
• Salma Kuhlmann
• Vincenzo Mantova
• Martin Hils
• Annalisa Conversano
##### 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.

##### Members:
• Mauro Di Nasso
• Moreno Pierobon
• Mariaclara Ragosta
##### Collaborators:
• Lorenzo Luperi Baglini (Università di Milano Statale)
• Isaac Goldbring (Irvine University)
• Renling Jin (Charleston University)
##### 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
• Marco Forti
##### Collaborators:
• Andreas Blass (Ann Arbor)
##### 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
• Marcello Mamino
##### Collaborators:
• Manuel Bodirsky (Dresden)
• Caterina Viola (Dresden)

#### Members

##### Staff
MauroDi Nasso[Mathscinet]
MarcelloMamino[Mathscinet]
##### Postdoctoral Fellows
RosarioMennuni[Mathscinet] [Orcid]
##### Ph.D. Students
MorenoPierobon
MariaclaraRagosta
##### External Collaborators
MarcoForti[Mathscinet]

#### Grants

• ##### Mathematical Logic: models, sets, computability (Prin 2017)

Principal Investigator: Alessandro Berarducci

Project period: 01/08/2019 - 19/02/2023

• ##### Sistemi dinamici in logica, geometria, fisica matematica e scienza delle costruzioni (Progetti di Ricerca di Ateneo (PRA) 2020 - 2021)

Coordinator of the Research Unit: Giacomo Tommei

Members of the Research Unit: Marco Abate, Riccardo Barsotti, Giulio Baù, Alessandro Berarducci, Gianluigi Del Magno, Mauro Di Nasso, Giovanni Federico Gronchi, Stefano Marò, Giacomo Lari, Daniele Serra, David Antonio Riccardo Mustaro

Project period: 07/07/2020 - 31/12/2022

• ##### Sistemi dinamici in analisi, geometria, logica e meccanica celeste (Progetti di Ricerca di Ateneo (PRA) 2017 - 2018)

Coordinator of the Research Unit: Marco Abate

Members of the Research Unit: Marco Abate, Claudio Bonanno, Mauro Di Nasso, Stefano Galatolo, Alessandro Berarducci, Giovanni Federico Gronchi, Carlo Carminati, Giacomo Tommei

Project period: 10/04/2017 - 09/04/2019