Oberseminar mathematische Logik

Advanced talks on mathematical logic by guests and members of the logic group

Organized by

• Prof. Dr. Peter Koepke
• Prof. Dr. Stefan Geschke
• Dr. Philipp Schlicht

Time and location

Mondays 16:30-18:00 at Endenicher Allee 60, room 1.007.
The members of the seminar are welcome for coffee and tea at the Hausdorff-Raum 009 (this is the "old" room number above the doorframe; it's next to 1.011) from 16:00-16:30 before the talks.

Programme

19 October Laurenţiu Leuştean (Technische Universität Darmstadt and Mathematisches Forschungsinstitut Oberwolfach) Proof mining in metric fixed point theory and ergodic theory

By proof mining we mean the logical analysis, using proof-theoretic tools, of mathematical proofs with the aim of extracting relevant information hidden in the proofs. This new information can be both of quantitative nature, such as algorithms and effective bounds, as well as of qualitative nature, such as uniformities in the bounds or weakening the premises. Thus, even if one is not particularly interested in the numerical details of the bounds themselves, in many cases such explicit bounds immediately show the independence of the quantity in question from certain input data. This line of research has its roots in Kreisel's program on unwinding of proofs, put forward in the 50's.

In this talk we present applications of proof mining in metric fixed point theory and ergodic theory:
- effective results on important classes of iterations associated to nonexpansive mappings;
- an explicit uniform bound on the metastability of ergodic averages in uniformly convex Banach spaces.

26 October Martin Ziegler (Universität Freiburg) Low function of reals

(Gemeinsame Arbeit mit Katrin Tent). Wir f�hren einen neuen Begriff der Berechenbarkeit von reellen Funktionen ein und beweisen einige Eigenschaften. Es folgt zum Beispiel, da� die Zetafunktion auf jeder beschr�nkten offen Teilmenge der komplexen Eben low-elementary ist. Es ergibt ich au�erdem ein vereinfachter Beweis des Satzes von Yoshinaga, der aussagt, da� alle Perioden von Kontsevich und Zagier low-elementary sind.

06 November Friday 14-16 at room 0.011 Benedikt Löwe (Universiteit Amsterdam) The modal logic of forcing

In set theory, forcing is one of the few fundamental techniques to build new models of set theory. It is therefore interesting to understand what type of results can be obtained by means of forcing.

In the spirit of Solovay's famous results about the modal logic of provability, we consider the "modal logic of forcing" ---the collection of truths about "being obtainable by the method of forcing" in a simple modal language--- and show that it coincides with a well-known modal logic, called S4.2 (the modal logic of reflexive, transitive, and directed frames).

In this talk reporting on joint work with Joel Hamkins (CUNY), I discuss the proof of our main theorem and discuss the relation to the multiverse view of set theory.

09 November Daniel Herden (Universität Duisburg-Essen and Universität Münster) Constructing absolute E-rings

A ring R with 1 is called an E-ring if End(R^+) is ring-isomorphic to R under the canonical evaluation map. After a short introduction to E-rings we will focus on the construction of absolute E-rings, i.e. on rings that remain E-rings in any generic extension. We will show that the first omega-Erdös cardinal kappa is a natural bound for the existence of such absolute objects: We give proof that there are no absolute E-rings of cardinality at least kappa and we will outline a construction for absolute E-rings of any cardinality below kappa using a suitable family of absolutely rigid colored trees.

23 November Ali Enayat (American University, Washington DC) NFU, 40 years later

In 1969 Jensen introduced the variant NFU of Quine's NF and proved the consistency of NFU relative to a fragment of Zermelo set theory. In this talk I will provide an overview of various unexpected meta-mathematical links between NFU-style set theories and ''orthodox'' foundational systems that have come to light since Jensen's pioneering work. As we shall see, NFU and its natural extensions are closely related to a range of ''orthodox'' foundational systems, ranging from exponential function arithmetic, all the way up to Kelley-Morse theory of classes with a weakly compact class of ordinals.

30 November Jeremy Avigad (Carnegie Mellon University and INRIA-Microsoft Research Joint Centre, Orsay) A formal system for Euclidean diagrammatic reasoning

This talk presents work carried out jointly with Ed Dean and John Mumma.

For more than two thousand years, Euclid's Elements was viewed as the paradigm of rigorous argumentation. But this changed in the nineteenth century, with concerns over the use of diagrammatic inferences and their ability to secure general validity. Axiomatizations by Pasch, Hilbert, and later Tarski are now taken to rectify these shortcomings, but proofs in these axiomatic systems look very different from Euclid's.

In this talk, I will argue that proofs in the Elements, taken at face value, can be understood in formal terms. I will describe a formal system with both diagram- and text-based inferences that provides a much more faithful representation of Euclidean reasoning. For the class of theorems that can be expressed in the language, the system is sound and complete with respect to Euclidean fields, that is, the semantics corresponding to ruler and compass constructions.

The system's one-step inferences are smoothly verified by current automated reasoning technology. This makes it possible to formally verify Euclidean diagrammatic proofs, and provides useful insight into the nature of mathematical proof more generally.

11 January Stefan Geschke (Universität Bonn) Weak Borel chromatic numbers

Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G-independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge. We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most aleph_1. Slightly weaker results hold for F_sigma -graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable F_sigma-graph has a Borel chromatic number of at most aleph_1.

18 January Peter Koepke (Bonn) Violating SCH without large cardinals

We present a symmetric forcing construction where a model $M$ of ZFC + GCH is extended to a model $N$ of ZF + "GCH holds below $\aleph_\omega$" + "there is a surjection from the power set of $\aleph_\omega$ onto $\theta$" where $\theta$ is an arbitrarily high fixed cardinal. First add $\aleph_n$ many subsets of $\aleph_n$ for every $n$. Then adjoin $\theta$ many $\omega$-sequences of the new subsets. The model $N$ is some choiceless submodel generated by $\theta$ many equivalence classes of subsets of $\aleph_\omega$ modulo some restricted variations. This is joint work with Moti Gitik which was inspired by earlier joint work with Arthur Apter.

01 February Vladimir Kanovey (Moscow) Borel reducibility as an additive property of domains

We prove that under certain requirements if E and F are Borel equivalence relations, X is a countable union of Borel sets X_n, and the restricted relation E | X_n is Borel reducible to F for all n, then E | X itself is Borel reducible to F. Thus the property of Borel reducibility to F is countably additive as a property of domains.

08 February Menachem Kojman (Ben Gurion University) From Hausdorff's gap to Shelah's scales