**Lecture series on Logic and Foundations of Mathematics (6)**

**Lecture Title:**** Kripke-Platek set theory**

**Speaker: **Dr. Fedor Pakhomov (Steklov Mathematical Institute of Russian Academy of Sciences, Russia)

**Date and Time: **2020-12-01, 7pm-9pm Beijing time (11am-1pm UTC)

**Platform:** Zoom (ID: 611 4696 9436)

**Organizer: **School of Philosophy, Wuhan University

**Interlocutor: **Dr. Zachiri McKenzie (School of Mathematics, University of Technology, Australia)

**Host: **Prof. Yong Cheng (Wuhan University)

**Abstract:**

Kripke-Platek set theory is a relatively weak set theory that has multiple connections with the fields of computability theory, proof-theory and model theory. In this talk I will introduce the theory and outline some of the connections.

**About the speaker:**

Dr. Fedor Pakhomov is a research fellow at Steklov Mathematical Institute of Russian Academy of Sciences in Russia. He got PhD in mathematics from Steklov Mathematical Institute in Russia. After that, he got postdoc positions at Institute of Mathematics of the Czech Academy of Sciences in Czech Republic, and School of Mathematics at Ghent University in Belgium. His research interests are mainly in the fields of proof theory and foundations of mathematics. His recent research work covers ordinal analysis, provability logics, limits of applicability of Goedel's 2nd incompleteness theorem, formal theories of truth, and Presburger arithmetic.

**Lecture series on Logic and Foundations of Mathematics (7)**

**Lecture Title: ****How many real numbers are there?**

**Speaker: **Prof.Ralf Schindler (University of Muenster, Germany)

**Date and Time: **2020-12-08, 16:00-18:00 (8am-10am UTC)

**Platform: **Zoom (ID: 655 7906 2464)

**Organizers:**

Tianyuan Mathematical Center in Central China

School of Mathematics and Statistics, Wuhan University

School of Philosophy, Wuhan University

**Interlocutors: **

Prof. Qi Feng (Chinese Academy of Sciences), Dr. Liuzhen Wu (Chinese Academy of Sciences), Dr. Guozhen Shen (Wuhan University)

**Host: **Prof. Yong Cheng (Wuhan University)

**Abstract:**

Georg Cantor showed that there are uncountably many real numbers. But how many of them are there? Two sets of axioms have been intensively studied over the last 30 years which both imply that there are exactly ℵ_{2} many real numbers: Forcing axioms and the P_{max} axiom (*), but until recently they seemed to be competitors. Both of them encapsulate the idea that the universe of sets is rich or “saturated” in a precise way. Last year, David Asperó and myself verified that forcing axioms and (*) are actually compatible, a fact that might support the view that the right axiom to decide how many real numbers there are has already been found.

**About the speaker:**

Ralf Schindler is a full professor in the department of mathematics and computer science at University of Muenster, and was the vice chair of the department. He was also Guest professor at UC Berkeley and the University of Barcelona. He was the co-editor of the Journal of Symbolic Logic and is the Editor-in-chief of the Archive for Mathematical Logic. He is the secretary of the European Set Theory Society and has served in various committees of international conferences and workshops. Ralf Schindler is an expert in set theory. His research interest in set theory includes inner model theory, large cardinals, descriptive set theory, and forcing axioms. He has made many contributions in set theory. Last year, together with David Asperó he answered a long standing open question in pure set theory by showing that Martin's Maximum^{++} implies Woodin's P_{max} axiom (*), which has strong impact on Cantor's Continuum Problem which asks how many real numbers there are.

**Lecture series on Logic and Foundations of Mathematics (8)**

**Title****: Kurt Gödel and Alfred Tarski:The Extremes of Logic**

**Speaker: **Prof. Matthias Baaz (Vienna University of Technology)

**Date and Time: **2021-01-14, 19:30-21:30

**Platform: **Zoom (ID: 627 9053 8435, Password: 877374)

**Organizer:** School of Philosophy, Wuhan University

**Host: **Prof. Yong Cheng (Wuhan University)

**Abstract:**

In this lecture commemorating the birth of Alfred Tarski and the death of Kurt Gödel, we will compare the eminent founders of modern logic: Kurt Gödel and Alfred Tarski.

Kurt Gödel has been driven by the the possibility that the individual thinking might transgress its own limits. The solutions of mathematical problems are the models, not the aims of his thinking. He strongly believed in the simplicity of all solutions maybe beyond language. Therefore he chose very carefully the next generation scientists with whom he communicated (basically Georg Kreisel, Gaisi Takeuti and Hao Wang).

Alfred Tarski on the other hand grew up in the logical traditions of Poland. He considered logic as a mathematical subject based on a mathematical language, which is very able to contribute to mathematics as algebra, topology etc. He emphasized the formal semantical relations as entailment, satisfaction and truth. He educated many students and influenced not only logic and mathematics but also formal linguistics by his thorough mathematical rigor.

**About the speaker:**

Matthias Baaz is a world-renowned logician. He studied Mathematical Logic and Astronomy at the University of Vienna and changed to the Vienna University of Technology where he currently serves as chair of computational logic within the Faculty of Mathematics. The main research field of Matthias Baaz is proof theory, but he works also in Gödel logics and Automated Theorem Proving. He has made many contributions in these fields. His main scientific advisor for most of his scientific life has been Georg Kreisel, one of the only three logicians of the next generation Kurt Gödel had an intensive communication with. Matthias Baaz is the Executive Vice President of the Kurt Gödel Society. He has organized many conferences for the Kurt Gödel Society, the largest event being the Vienna Summer of Logic 2014 with more than 2500 participants.

**Lecture series on Logic and Foundations of Mathematics (9)**

**Lecture Title****: Reflection Algebras and Progressions**

**Speaker: **Prof. Lev D. Beklemishev (Steklov Institute of Mathematics, Russian Academy of Sciences)

**Date and Time: **2021-01-21, 16:00-18:00 Beijing time (UTC 8:00-10:00)

**Platform: **Zoom (ID: 683 7699 3624, password: 919183)

**Organizer: **School of Philosophy, Wuhan University

**Host: **Prof. Yong Cheng (School of Philosophy, Wuhan University,)

**Abstract:**

Reflection principles are axioms expressing that all sentences (of a given logical complexity) provable in a given theory *T *are true. The simplest example of such an axiom is Goedel’s formula expressing the consistency of *T*. The idea of using reflection principles and their transfinite iterations to classify arithmetical sentences according to strength is due to A. Turing (1939). However, Turing also realized that there are serious difficulties associated with this approach, in particular, due to the lack of understanding how to distinguish ‘canonical’ from ‘pathological’ ordinal notation systems, now a well-known problem in proof theory. The aim of the talk is to outline the main ingredients of the approach to proof theoretic analysis based on reflection algebras. From an abstract algebraic point of view, these structures are semi-lattices enriched by a family of monotone unary operators satisfying some specific sets of identities. The operators can be interpreted in the lattice of arithmetical theories as functions mapping a theory *T *to a theory axiomatized by a reflection principle for *T*. Within this framework it is possible to define appropriate canonical ordinal notation systems and the associated transfinite hierarchies of reflection principles. Hence, it is also possible, to some extent, to push Turing’s ideas on the classification of arithmetical sentences through.

**About the speaker:**

Prof. Lev D. Beklemishev is a world-renowned logician, an academician of Russian Academy of Sciences, the deputy director of Steklov Mathematical Institute in Moscow, the chair of the Department of mathematical Logic, and the deputy director of Steklov International Mathematical Center. He has made many contributions in logic and his main areas of research include proof theory, provability logic, modal and non-classical logic, reflection principles and progressions of axiomatic systems, incompleteness, and logic for Computer Science. He has organized many international workshops and conferences. He is the editor of The Journal of Symbolic Logic, and was Executive Council member of Association for Symbolic Logic (ASL).