## on Weak Arithmetics and Set Theories

#### Objective

The primary objective of this laboratory is to establish a mean for collaborative efforts between all the Partners. The goal is to promote common research activities and mobility opportunities for Master and PhD students, postdoctoral fellows and faculty members one the topics of weak arithmetics and set theories. One of its goals is to help into the organization of the JAF yearly conference on weak artithmetics.

#### Research Topics of interest

The field of *Weak arithmetics* is application of logical methods to Number Theory. The most famous results are: **undecidability of elementary arithmetic** (in contrast with elementary geometry, there is no computer program to determine whether an arithmetical sentence is true); and the negative answer to the Hilbert’s tenth problem (there is no general algorithm to decide whether a diophantine equation with integer

coefficients has a solution in integers).

Number Theory is free to use any method to obtain results concerning natural integers. The adjective weak in weak arithmetics refers to restrictions used in this topic. First of all, weak arithmetics specify its object of study: it is a study of the first order structure ⟨ℕ,+,×⟩. In fact, we study expansions by definitions of this structure and substructures of such expansions. Because ℕ is not well defined in the universe of sets, we also study **non standard models** of ⟨ℕ,+,×⟩, more precisely of Th(ℕ,+,×), where Th means *theory* in a formal sense. As well as we are interested in nonstandard models of Th(A), where A is a substructure of an expansion by definitions of ⟨ℕ,+,×⟩ or Th(A) is a theory given by a set of axioms. The same can be done for the structure ⟨V,∈⟩ where V is the universe of **set theory**.

Secondly, weak arithmetics and weak set theories also impose a r**estriction on the studied properties**. We do not consider ill defined properties but only well defined logical sentences: first-order sentences and second-order sentences. The origin of this restriction comes from set theory. After the axiomatization of set theory by Zermelo, Henri Poincaré, criticized the explanation given by Zermelo of the notion of *defined property* implied in the axiom of separation. In 1922, Fraenkel and Skolem proposed, independently, a more precise definition and nowadays we prefer to use the second one: to employ a formal first-order language and to consider a defined property to be a property that can be expressed by a first-order formula, called a definable property.

#### Events

21-10-21: Meeting in Athens for the creation of an international master

28-06-22: Research days in honor of Patrick Cegielski

#### Consortium

- Laboratoire Algorithmique, Complexité et Logique at the Université Paris-Est Créteil Val-de-Marne
- Department of History and Philosophy of Science at the University of Athens
- Centro de Matemática, Aplicações Fundamentais e lnvestigação Operacional at the Faculdade de Ciências da Universidade de Lisboa
- Departamento Ciencias de la Computación e Inteligencia Artificial at the Universidad de Sevilla

#### Members

- Patrick Cegielski (Paris)
- Julien Cervelle (Paris)
- Andrès Cordon Franco (Sevilla)
- Fernando Ferreira (Lisbon)
- Gilda Ferreira (Lisbon)
- Patros Stefanea (Athens)
- Yanis Stephanou (Athens)
- Pierre Valarcher (Paris)