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Alessandro Berarducci

Publications and source records attributed to Alessandro Berarducci.

At least 19 recordsLinked to original sources
arXiv2026

Projective curves and weak second-order logic

Given an algebraically closed field $K$ of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of $\mathbb P^2(K)$. Answering a question of Marcus Tressl, we prove that the poset interprets the field, and it is in fact bi-interpretable with the two-sorted structure consisting of the field $K$ and a sort for its finite subsets. In this structure one can define the integers, so the theory is undecidable. When $K$ is the field of complex numbers we can nevertheless obtain a recursive axiomatization modulo the theory of the integers. We also show that the integers are stably embedded and that the poset of irreducible varieties over the complex numbers is not elementarily equivalent to the one over the algebraic numbers.

math.LO↗
arXiv2026

On the elementary theory of the real exponential field

Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre and Wilkie that, under the same conjecture, $T_{\exp}$ is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to $(-1,1)$, which we prove unconditionally.

math.LO↗
arXiv2023

Vector spaces with a union of independent subspaces

Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and admits quantifier elimination in the language of $K$-vector spaces with predicates for the $n$-fold sums of $X$ with itself. If $K$ is finite this is no longer true, but we still have that a natural completion is near-model-complete.

math.LO↗
arXiv2022

Orthogonal decomposition of definable groups

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. Finally, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.

math.LO↗
arXiv2022

Provability Logic: models within models in Peano Arithmetic

In 1994 Jech gave a model theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem, that the existence of a model of PA of complexity $Σ^0_2$ is independent of PA, where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic with the modal operator interpreted as truth in every $Σ^0_2$-model.

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arXiv2021

Is the twin prime conjecture independent of Peano Arithmetic?

We show that there is an arithmetical formula F such that ZF proves that F is independent of PA and yet, unlike other arithmetical independent statements, the truth value of F cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula F such that ZF proves that F is equivalent to the twin prime conjecture.

math.LO↗
arXiv2020

Asymptotic analysis of Skolem's exponential functions

Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant $1$, the identity function $x$, and such that whenever $f$ and $g$ are in the set, $f+g,fg$ and $f^g$ are in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below $2^{2^x}$. Here we prove that the set of asymptotic classes within any archimedean class of Skolem functions has order type $ω$. As a consequence we obtain, for each positive integer $n$, an upper bound for the fragment below $2^{n^x}$. We deduce an epsilon-zero upper bound for the fragment below $2^{x^x}$, improving the previous epsilon-omega bound by Levitz (1978). A novel feature of our approach is the use of Conway's surreal number for asymptotic calculations.

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arXiv2019

Exponential fields and Conway's omega-map

Inspired by Conway's surreal numbers, we study real closed fields whose value group is isomorphic to the additive reduct of the field. We call such fields omega-fields and we prove that any omega-field of bounded Hahn series with real coefficients admits an exponential function making it into a model of the theory of the real exponential field. We also consider relative versions with more general coefficient fields.

math.LO↗
arXiv2017

Transseries as germs of surreal functions

We show that Écalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.

math.LO↗
arXiv2017

Cartan subgroups and regular points of o-minimal groups

Let G be a group definable in an o-minimal structure M. We prove that the union of the Cartan subgroups of G is a dense subset of G. When M is an expansion of a real closed field we give a characterization of Cartan subgroups of G via their Lie algebras which allow us to prove firstly, that every Cartan subalgebra of the Lie algebra of G is the Lie algebra of a definable subgroup - a Cartan subgroup of G -, and secondly, that the set of regular points of G - a dense subset of G - is formed by points which belong to a unique Cartan subgroup of G.

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arXiv2017

A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets

Results of Smale (1957) and Dugundji (1969) allow to compare the homotopy groups of two topological spaces $X$ and $Y$ whenever a map $f:X\to Y$ with strong connectivity conditions on the fibers is given. We apply similar techniques in o-minimal expansions of fields to compare the o-minimal homotopy of a definable set $X$ with the homotopy of some of its bounded hyperdefinable quotients $X/E$. Under suitable assumption, we show that $π_{n}(X)^{\rm def}\congπ_{n}(X/E)$ and $\dim(X)=\dim_{\mathbb R}(X/E)$. As a special case, given a definably compact group, we obtain a new proof of Pillay's group conjecture "$\dim(G)=\dim_{\mathbb R}(G/G^{00}$)" largely independent of the group structure of $G$. We also obtain different proofs of various comparison results between classical and o-minimal homotopy.

math.LO↗
arXiv2015

Surreal numbers, derivations and transseries

Several authors have conjectured that Conway's field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.

math.LO↗
arXiv2013

Groups definable in two orthogonal sorts

This work can be thought as a contribution to the model theory of group extensions. We study the groups G which are interpretable in the disjoint union of two structures (seen as a two-sorted structure). We show that if one of the two structures is superstable of finite Lascar rank and the Lascar rank is definable, then G is an extension of a group internal to the (possibly) unstable sort by a definable subgroup internal to the stable sort. In the final part of the paper we show that if the unstable sort is an o-minimal expansion of the reals, then G has a natural Lie structure and the extension is a topological cover.

math.LO↗
arXiv2012

Discrete subgroups of locally definable groups

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group G in this category is a cover of a definable group. We prove that this is the case under a natural convexity assumption inspired by the same authors, which in fact gives a necessary and sufficient condition. The proof is based on the study of the zero-dimensional compatible subgroups of G. Given a locally definable connected group G (not necessarily definably generated), we prove that the n-torsion subgroup of G is finite and that every zero-dimensional compatible subgroup of G has finite rank. Under a convexity hypothesis we show that every zero-dimensional compatible subgroup of G is finitely generated.

math.LO↗
arXiv2011

Topology of definable abelian groups in o-minimal structures

In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of the same dimension. Moreover, in the semialgebraic case the result holds for all dimensions.

math.LO↗
arXiv2010

Group covers, o-minimality, and categoricity

We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case of covers of complex algebraic groups studied by Zilber and his students. We also discuss from a model-theoretic point of view the following question, related to "Milnor's conjecture": is a finite central extension (as an abstract group) of a compact Lie group also a topological extension?

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