2016 Bulletin of Symbolic Logic  
Weiermann (Ghent University). The main conference topics were: Model Theory, Set Theory, Computability Theory, Proof Theory, Philosophy of Mathematics and Logic, Logic and Quantum Foundations. The program included two tutorial courses, eleven invited plenary lectures, and twenty-four invited lectures in six special sessions. There were 167 contributed papers, of which 7 were by title only, and in all 222 participants from many different countries. Twenty-one students and recent Ph.D.'s were
more » ... ded ASL grants. The joint CLMPS/LC2015 opening plenary lecture was given by Johan van Benthem (University of Amsterdam and Stanford University), with the title: Logic in play. The following tutorial courses were given: Erich Grädel (University of Aachen), Dependence and independence in logic. Menachem Magidor (Hebrew University, Jerusalem), On the set theory of generalized logics. The following invited plenary lectures were presented: Toshiyasu Arai (Chiba University), Proof theory of set theories. Sergei Artemov (CUNY), Constructive knowledge. (joint with CLMPS) Steve Awodey (University of Pittsburgh), Cubical homotopy type theory and univalence. Dependence and independence are general scientific concepts that play a fundamental role in many disciplines. Classical proposals for incorporating concepts of dependence or independence into mathematical logic include Henkin quantifiers and independence friendly logics giving for each quantifier explicit information on how variables may or may not depend on each other. Modern dependence logics instead treat dependence and independence as atomic properties, rather than as annotations of quantifiers. Dependence and independence are concepts that do not manifest themselves in a single assignment, mapping variables to elements of a structure, but only in larger amounts a data, such as a table or relation, or a set of assignments. Accordingly, logics of dependence and independence have a semantics that, unlike Tarski semantics, are based on sets of assignments. Sets of assignments are called teams and the semantics is called team semantics. In this tutorial we shall introduce a number of variations of logics of dependence, discuss their model-theoretic properties, expressive power, and algorithmic complexity. We shall design model-checking games for logics with team semantics in a general and systematic way, based on a notion of second-order reachability games. One of the most intriguing results on logics of dependence and independence is the tight connection between inclusion logic and the least fixed-point logic LFP. We shall discuss this connection from a game-theoretic point of view, by showing that the evaluation problems for both logics can be represented by a special kind of trap condition in safety games. We then study interpretation arguments for games that provide a model-theoretic construction of translations between these logics. MENACHEM MAGIDOR, On the set theory of generalized logics. Many interesting properties of mathematical structures and elements of such structures can not be expressed in first order logic. Generalized logics attempt to provide mechanisms for expressing such properties. Second order logic is a prime example of such logic. When one attempts to develop the model theory of a logic generalizing first order logic, one runs into the problem that model theoretic properties of the logic like compactness, Löwenheim-Skolem Theorem, the robustness of the set of validities of the logic across different universes of Set Theory etc. depend very much on the set theoretical assumptions at https://www.cambridge.org/core/terms. https://doi.Proof theory is a branch of mathematical logic in which (mathematical) proofs are treated as formal objects, while in (axiomatic) set theory we investigate universes of sets and propositions which are supposed to hold in universes of sets. Universes of sets are ought to obey axioms of (formal) set theory. Here we are concerned with formal proofs in set theory asking questions: Which kind of sets are proved to exist? In set theory, a reflection principle says that a proposition true in the universe holds already in a smaller set. The reflection principle has been one of sources to formulate large cardinals, e.g., indescribable cardinals due to Hanf-Scott. A hierarchy of indescribable cardinals is known to be obtained by enlarging classes of formulas to be reflected, and/or restricting reflecting points. On the other side, recursive analogues of small large cardinals such as indescribable cardinals (reflecting ordinals due to Richter-Aczel) have been investigated in proof theory, i.e., ordinal analyses of extensions of Kripke-Platek set theory. Combining these two, we report recent progress in proof-theoretic investigations of set theory such as bounding on provably existing countable ordinals and proof-theoretic reductions of higher indescribability to iterations of lower indescribabilities. SERGEI ARTEMOV, Constructive knowledge.
doi:10.1017/bsl.2016.22 fatcat:cm4a5yyvgfajzm4qvlszviblni