The summer school is addressed at students of mathematics, philosophy, computer science and other subjects that are related to mathematics with basic knowledge in logics. Preference will be given to undergraduate students in their final year and graduate students. The summer school conveys insight into current research topics in the areas of logics and computability. The scientific program consists of mini courses (with lectures, exercises and projects). English is the teaching language.
- Gödel's Incompleteness Theorems (Reinhard Kahle, Tübingen)
Gödel's Incompleteness Theorems make part of the general mathematical culture. They concern the unprovability of certain formal statements in axiomatic theories. Their understanding, however, is often clouded by philosophical and technical inaccuracies. The aim of this course is to give a concise introduction to these results, their proofs and their relevance. We will, in particular, focus on the fundamental distinction between syntax and semantic.
- The Axioms of Zermelo and Fraenkel (Heike Mildenberger, Freiburg)
Whereas our concepts on finite sets and strings and unambiguous readability can be justified from everyday experience, our concepts on infinite sets are based on cultural conventions. Mathematics is based on axioms and on proof rules. Most mathematicians get acquainted to the mainstream version tacitly or explicitly in their first courses with rigorous proofs.
Especially infinitary combinatorics depends heavily on the underlying axioms and proof rules. These lead to desired consequences and inevitable by-products: E.g., if every vector space has a base, then there is a subset of the real line that is not Lebesgue measurable. The course will focus on some combinatorial consequences of the Zermelo-Fraenkel-Axioms.
- Recursion and Complexity (Isabel Oitavom, Lisbon)
Computability theory provides us with theoretical results about what can be computed. However, just a fraction of the theoretically computable functions can be implemented through efficient algorithms. Complexity theory addresses this question, usually using machine models, like the Turing machine, and explicit bounds. In contrast, implicit characterizations of complexity classes avoid both machines and explicit complexity conditions and therefore allow a more mathematically description of complexity classes. In this minicourse different characterisations of complexity classes based on recursive schemes will be discussed.
- Connexive logic (Hitoshi Omori & Heinrich Wansing, Bochum)
Most prominent non-classic logics are subsystems of classic logic. This is different with connexive logics (see https://plato.stanford.edu/entries/logic-connexive/ for an overview). The characteristic feature of systems of connexive logic lies in that principles such as Aristotle's Theses (~(~A→A), ~(A→~A)) and Boethius' Theses ((A→B)→~(A→~B), (A→~B)→~(A→B)), which are not theorems of classical logic, are included as theorems. In this sense, systems of connexive logic are contra-classical, and one of the major challenges of connexive logic is to define non-trivial but convincingly motivated logics in which connexive principles are valid/derivable. Motivation for systems of the connexive logic include considerations of the relevance of valid conclusions, considerations of understanding negation, and empirical studies of understanding negated conditionals in natural languages. The course offers a systematic overview of a rapidly developing area with a special emphasis on the system C which can be seen as a very natural expansion of intuitionistic logic.