
There will be a series of tutorials before the symposium. These tutorials will be held from May 30 to June 2. Location: Universidad Nacional (Schedule).
If you’re attending the tutorials, please fill out this registration form.
Model Theory: Recent developments in model theoryBy Hans Adler (Vienna University)
I will give an introduction to the ‘pure’ side of model theory of first order logic, which in some sense asks why some fields of mathematics are easier or more beautiful than others. (Shelah’s structure/nonstructure dichotomies.) Started with Michael Morley’s famous categoricity theorem and to a large extent developed singlehandedly by Saharon Shelah, the focus of the field has become ever more general, moving on from uncountably categorical theories via stable theories to simple and dependent theories. A hot topic of fundamental research at the moment is the huge class of theories without the tree property of the second kind.
Set Theory: Determinacy and inner model theoryBy Andrés Eduardo Caicedo (Boise State University)
Since the invention of forcing, we know of many statements that are independent of the usual axioms of set theory, and even more that we know are consistent with the axioms (but we do not yet know whether they are
actually provable).These proofs of consistency typically make use of large cardinal assumptions. Inner model theory is the most powerful technique we have developed to analyze the structure of large cardinals. It also allows us to show that the use of large cardinals is in many cases indispensable. For years, the main tool in the development of this area was fine structure theory.
Determinacy (in suitable inner models) is a consequence of large cardinals, and recent work has revealed deep interconnections between determinacy assumptions and the existence of inner models with large
cardinals, thus showing that descriptive set theory is also a key tool.The development of these connections started in earnest with Woodin’s core model induction technique, and has led to what we now call Descriptive inner model theory.
The goal of the minicourse is to give a rough overview of these developments.
Algebraic Logic: Algebraic Proof TheoryBy Nikolaos Galatos (University of Denver)
Proof Theory, initiated by Gentzen’s seminal work on sequent calculi in the 1930′s, aims at analyzing the structure of proofs. The key result is the cutelimination theorem, which results in analytic systems (namely, no statement, like a lemma, or connective that is foreign to a theorem needs to be introduced in its proof). Analytic systems enjoy nice properties, often including decidability. Gentzen’s prooftheoretic methods also extend to (nonclassical) substructural logics, including intuitionistic, linear and relevance, to name a few. These logical systems have applications in computer science (lambda calculus and type theory), philosophy and linguistics and although their propositional fragments are not as expressive as firstorder classical logic they are decidable.
On a different theme, traditional algebraic logic, introduced mainly by Tarski, aims at using methods of (universal) algebra to study logic, for example via the study of cylindric algebras. The (universal) algebraic approach has proved useful in the study of substructural logic via the study of residuated lattices, algebraic structures that were originally defined in the context of the lattice of ideals of rings.
However, only recently have algebraic methods been used to study the prooftheoretic aspects of these logics. Algebraic proof theory aims at bringing the two areas of Proof Theory and Algebraic Logic together. It provides general and modular proofs of cutelimination theorems and serves at systematizing and extending traditional proof theory by algebraic and relationaltheoretic means.