# Set Theory and its Interactions

Organizadores: Carlos Di Prisco (ca.di@uniandes.edu.co), Christina Brech (brech@ime.usp.br)

• #### Lunes 13

15:00 - 15:45

#### Hereditary interval algebras and cardinal characteristics of the continuum

Carlos Martinez-Ranero (Universidad de Concepción, Chile)

An interval algebra is a Boolean algebra which is isomorphic to the algebra of finite unions of half-open intervals, of a linearly ordered set. An interval algebra is hereditary if every subalgebra is an interval algebra. We answer a question of M. Bekkali and S. Todorcevic, by showing that it is consistent that every $$\sigma$$-centered interval algebra of size $$\mathfrak{b}$$ is hereditary. We also show that there is, in ZFC, an hereditary interval algebra of cardinality $$\aleph_1$$.

15:45 - 16:30

#### Preservation of some covering properties by elementary submodels

Lucia Junqueira (Universidade de São Paulo, Brasil) joint with Robson A. Figueiredo and Rodrigo R. Carvalho

Given a topological space $$(X, \tau)$$ and an elementary submodel $$M$$, we can define the topological space $$X_M = (X\cap M, \tau _M)$$, where $$\tau _M$$ is the topology on $$X \cap M$$ generated by $$\{ V\cap M : V \in \tau \cap M \}$$. It is natural to ask which topological properties are preserved by this new operation. For instance, if $$X$$ is $$T_2$$, then $$X_M$$ is also $$T_2$$. On the other hand, $$X_M$$ compact implies $$X$$ compact. A systematic study of it was initiated by L. Junqueira and F. Tall in 1998.

In the paper More reflection in topology'', published in Fudamenta Mathematicae in 2003, F. Tall and L. Junqueira, studied the reflection of compactness and, more specifically, when can we have, for $$X$$ compact, $$X_M$$ compact non trivially, {\it i.e.}, with $$X \neq X_M$$. It is natural to try to extend this study for other covering properties.

We will present some results concerning the preservation of Lindelöfness. We will also discuss the perservation of some of its strengthenings, like the Menger and Rothberger properties.

16:45 - 17:30

#### Group operations and universal minimal flows

Dana Bartosova (University of Florida, Estados Unidos)

Every topological group admits a unique, up to isomorphism, universal minimal that maps onto every minimal (with respect to inclusion) flow. We study interactions between group operations and corresponding universal minimal flows.

17:30 - 18:15

#### Some lessons after the formalization of the ctm approach to forcing

Pedro Sánchez Terraf (Universidad Nacional de Córdoba, Argentina) joint with Emmanuel Gunther, Miguel Pagano, and Matías Steinberg

In this talk we'll discuss some highlights of our computer-verified proof of the construction, given a countable transitive set model $$M$$ of $$\mathit{ZFC}$$, of a generic extension $$M[G]$$ satisfying $$\mathit{ZFC}+\neg\mathit{CH}$$. In particular, we isolated a set $$\Delta$$ of $$\sim$$220 instances of the axiom schemes of Separation and Replacement and a function $$F$$ such that such that for any finite fragment $$\Phi\subseteq\mathit{ZFC}$$, $$F(\Phi)\subseteq\mathit{ZFC}$$ is also finite and if $$M\models F(\Phi) + \Delta$$ then $$M[G]\models \Phi + \neg \mathit{CH}$$. We also obtained the formulas yielded by the Forcing Definability Theorem explicitly.

To achieve this, we worked in the proof assistant Isabelle, basing our development on the theory Isabelle/ZF by L. Paulson and others.

The vantage point of the talk will be that of a mathematician but elements from the computer science perspective will be present. Perhaps some myths regarding what can effectively be done using proof assistants/checkers will be dispelled.

We'll also compare our formalization with the recent one by Jesse M. Han and Floris van Doorn in the proof assistant Lean.

• #### Martes 14

15:00 - 15:45

#### On non-classical models of ZFC

Giorgio Venturi (Universidade Estadual de Campinas, Brasil), joint work with Sourav Tarafder and Santiago Jockwich

In this talk we present recent developments in the study of non-classical models of ZFC.

We will show that there are algebras that are neither Boolean, nor Heyting, but that still give rise to models of ZFC. This result is obtained by using an algebra-valued construction similar to that of the Boolean-valued models. Specifically we will show the following theorem.

There is an algebra $$\mathbb{A}$$, whose underlying logic is neither classical, nor intuitionistic such that $$\mathbf{V}^{\mathbb{A}} \vDash$$ ZFC. Moreover, there are formulas in the pure language of set theory such that $$\mathbf{V}^{\mathbb{A}} \vDash \varphi \land \neg \varphi$$.

The above result is obtained by a suitable modification of the interpretation of equality and belongingness, which are classical equivalent to the standard ones, used in Boolean-valued constructions.

Towards the end of the talk we will present an application of these constructions, showing the independence of CH from non-classical set theories, together with a general preservation theorem of independence from the classical to the non-classical case.

15:45 - 16:30

#### The Katetov order on MAD families

Osvaldo Guzmán (Universidad Nacional Autónoma de México, México)

The Katetov order is a powerful tool for studying ideals on countable sets. It is specially interesting when restricted to the class of ideals generated by MAD families. One of the reasons we are interested in it is because it allows us to study the destructibility of MAD families under certain forcing extensions. In this talk, I will survey the main known results regarding the Katetov order on MAD families and state some open problems.

16:45 - 17:30

#### Around (*)

David Asperó (University of East Anglia, Inglaterra)

In this talk I will present work motivated by the derivation of the $$\mathbb P_{max}$$ axiom $$(*)$$ from Martin's Maximum$$^{++}$$.

17:30 - 18:15

#### Groups definable in partial differential fields with an automorphism

Samaria Montenegro (Universidad de Costa Rica, Costa Rica), joint work with Ronald Bustamente Medina and Zoé Chatzidakis

Model theory is a branch of mathematical logic with strong interactions with other branches of mathematics, including algebra, geometry and number theory.

In this talk we are interested in differential and difference fields from the model-theoretic point of view.

A differential field is a field with a set of commuting derivations and a difference-differential field is a differential field equipped with an automorphism which commutes with the derivations.

The model theoretic study of differential fields with one derivation, in characteristic $$0$$ started with the work of Abraham Robinson and of Lenore Blum. For several commuting derivations, Tracey McGrail showed that the theory of differential fields of characteristic zero with $$m$$ commuting derivations has a model companion called $$DCF$$. This theory is complete, $$\omega$$-stable and eliminates quantifiers and imaginaries.

In the case of difference-differential fields, Ronald Bustamante Medina (for the case of one derivation) and Omar León Sánchez (for the general case) showed that the theory of difference-differential fields with $$m$$ derivations admits a model companion called $$DCF_mA$$. This theory is model-complete, supersimple and eliminates imaginaries.

Cassidy studied definable groups in models of $$DCF$$, in particular she studied Zariski dense definable subgroups of simple algebraic groups and showed that they are isomorphic to the rational points of an algebraic group over some definable field.

In this talk we study groups definable in models of $$DCF_mA$$, and show an analogue of Phyllis Cassidy's result.