Let L be a first-order 2-sorted language. Let X be some fixed structure. A standard structure is an L-structure of the form ⟨M,X⟩. When X is a compact topological space (and L meets a few additional requirements) it is possible to adapt a significant part of model theory to the class of standard structures. In the last 20 years the most popular approach uses real-valued logic (Ben Yaacov, Berenstein, Henson, Usvyatsov).

In this course, we present a different, more general approach that only uses classical logic. This is based on three facts:

• Every standard structure has a positive elementary extension that is standard and realizes all positive types that are finitely consistent.
• In a sufficiently saturated structure, the negation of a positive formula is equivalent to an infinite disjunction of positive formulas.
• There is a pure model theoretic notion that corresponds to Cauchy completeness.

To exemplify how this setting applies to model theory, we discuss ω-categoricity and (local) stability. We will revisit the classical theory and compare it with the continuous case.

TENTATIVE PROGRAM

-PART 1: (ABSOLUTE) MODEL COMPANIONSHIP AND (A)MC-SPECTRA OF FIRST ORDER THEORY (6 to 10 hours): We give an extensive presentation of the basic results on model companionship and model completeness (e.g. see Chapter 3.5 of Chang-Keisler book or the notes: http://www.logicatorino.altervista.org/matteo_viale/MODCOMP.pdf). Then we introduce a slight strenghtnening of model companionship MC (absolute model companionship AMC) and we outline the main properties of this notion. Finally we introduce the notion of (A)MC-spectrum of a first order theory as a classification tool for such theories. Roughly the (A)MC-spectrum of a theory detects the signatures in which the theory can be first order axiomatized so to have a(n absolute) model companion. We outline why this is a non-trivial and informative classification tool for mathematical theories.

A basic knowledge of the standard results on first order logic (e.g. completeness and soundness theorem, compactness, Lowenheim-Skolem results) suffices for this part.

-PART 2: BASIC RESULTS ON THE (A)MC-SPECTRUM OF SET THEORY (6 to 10 hours): We show that if one considers as standard signatures for set theory only those which include predicate symbols for all \Delta_0-properties, and function symbols for all Goedel operations, then the (A)MC-spectrum of set theory includes many of these signature and the model companion of set theory with respect to these signatures describe structures resembling an H(\kappa^+) e.g structures which satisfy ZFC^- (e.g. ZFC minus the powerset axiom) and the axiom "every set has size at most \kappa" for \kappa a cardinal which depends on the signature. We also show that: (i) in any such signature L as above which includes a constant for the first uncountable cardinal, if T extends ZFC as axiomatized in L and T+\neg CH is consistent, CH cannot be in the absolute model companion of T for L. (ii) the same result applies for the natural formalization of 2^\omega>\omega_2 in the signature.

The set theory prerequisites for this part are familiarity with Mostowski collapsing theorem, and with the notion of "absolute concept" for set theory (e.g. Chapters I,III,IV of Kunen's textbook are more than sufficient)

PART 3: ADVANCED RESULTS ON THE AMC-SPECTRUM (10 hours or more): We outline a proof that there is at least one signature L* (as required in PART 2) such that any T extending ZFC+there are class many supercompact cardinals as axiomatized in signature L* admits an AMC and this AMC describes a theory of H(\aleph_2) in models of strong forcing axioms, in which a definable version of 2^\omega=\omega_2 holds. Furthermore this AMC is axiomatized by the \Pi_2-sentences for L* which are provably forcible over any model of T.

Note that the results apply even if T models CH or 2^\omega>\omega_2.

We do not know how much of this proof we would be able to give. However we should be able to give complete proofs of slightly weaker results.

This part will need familiarity with basic results on Woodin's stationary tower forcing and with basic facts on the Pmax technology.

Appropriate references for everything presented in the course will be given later.

Here is a tentative for a program:

• The first part of the course will focus on Ramsey-like results and their applications to Banach spaces. Ramsey results will include at least the Galvin-Prikry theorem, and transfinite combinatorial Ramsey results like the Nash-Williams theorem. Applications to Banach space theory will be for instance Rosenthal's characterizations of Banach spaces containing: on one hand $l^1$, and on the other $c_0$.

• In a second part of the course we will explore results binding Banach spaces to the first Baire class, like for instance the characterization of those spaces containing $l^1$ using the double-dual space by Odell-Rosenthal, or the Bourgain-Fremlin-Talagrand theorem on compact subsets of the first Baire class.

• If time allows, in a third part we shall follow more recent works, to choose between for instance Todorcevic's topological Ramsey results and their applications, or Gowers' strategical Ramsey results and their applications.

A finite set of minimal elements in a class of objects, for a given quasi-order, is the typical example of a finite basis. Many important results, notably in descriptive set theory, consist in giving the existence of a finite basis. The objective of this series of lectures is to first see some examples of such results, then to introduce the quasi-orders which always admit finite bases, also called well-quasi-orders, and their theory. We will also discuss various ways to prove the impossibility of a finite basis result.

• Metric structures. Ultraproducts of metric structures and compactness. Keisler-Shelah theorem for metric structures.
• Topologies on classes of structures. Model-theoretic forcing and Baire category. Omitting types and Cech completeness.
• Tao’s metastability. The Uniform Metastability Principle and connections with compactness.
• Type definability. Connections with combinatorics and the Grothendieck-Ptak double limit theorems.

Generalized Baire and Cantor spaces: definition and basic properties. Generalized Borel and analytic sets. Codings for uncountable structures, nonseparable spaces and alike. Some classification problems whose complexity can be settled in this framework. The regular case. Links with stability from model theory. The singular case: $\lambda$-Polish spaces.

The teaching material here linked is a draft of a book that will serve as the main reference for the course. I plan to cover in detail chapter 1, then depending on the background of the people attending I may just skim through chapters 3 and 4 or spend more time on them. Then I will make with great care chapters 6 and 7. If time remains we could either try to look at Woodin’s generic absoluteness results for second order arithmetic (chapters 2, 5, 12) or at the iteration theorem for semiproper forcings (chapters 2, 9, 10, and maybe parts of 8 or 11).
As a reference for the background material, I suppose the audience should be familiar with this: link

Con la tesi di dottorato di William W. Wadge vediamo la nascita della riduzione di Wadge. In generale, potremmo dire che riguarda la riduzione continua su spazi Polacchi. Una cosa davvero interessante di questa teoria e la sua rappresentazione come gioco sullo spazio di Baire. In questo seminario introdurremo questa teoria proprio partendo dal gioco di Wadge e per nire vedremo alcune applicazioni e risultati recenti.

Seminario Math-Lab - Per il riconoscimento vedi pagina Moodle.

Besides being a standard tool in set theory, ultrafilters are used in several different branches of mathematics, such as functional analysis or combinatorial number theory. This talk will focus on the applications of ultrafilters in topological dynamics, following the seminal works of Blass, Bergelson, Hindman and Strauss. After a brief introduction to the theory of ultrafilters, we will see the dynamical notion of recurrence in terms of ultrafilters, obtaining an algebraic description of some dynamical properties. Moreover, this way of looking at recurrence will reveal the link between dynamical notions and Ramsey-like properties of sets of natural numbers.

Seminario Math-Lab - Per il riconoscimento vedi pagina Moodle.

Model theory is a branch of mathematical logic with strong connections to most areas of mathematics and theoretical computer science. The last few decades have seen very interesting interactions between model theory and various classes of groups, in particular topological groups and Lie groups. Lie groups are smooth manifolds with a compatible group operation, and they are found everywhere in mathematics and all areas of science. If you have ever encountered a group of matrices during your studies, it was most likely a Lie group.

In this talk recent work on the connections between groups and model theory will be explained, and many examples will be presented, with a view towards future developments.

Zoom Meeting ID: 886 2603 1968

(*) La riunione si apre alle ore 9:20 con una chiacchierata informale in italiano. I seminario vero e proprio, in inglese, comincia alle 9:30 e durerà circa un'ora.

Seminario Math-Lab - Per il riconoscimento vedi pagina Moodle.