Tuesday, January 17, 2017, 15.00
Howard House 4th Floor Seminar Room
Speaker: Philip Welch (University of Bristol)
Title: The Ramified Analytical Hierarchy and Strong Logics
The ramified analytical hierarchy defined by Kleene builds up a hierarchy of models of subsystems of analysis in a second order definable manner.
We address a question of Kennedy as to what can be done using strong logics to re-define the stages of Kleene’s hierarchy, in the spirit of “Inner Models from Extended Logics” of Kennedy, Magidor, & Väänänen. In this paper they followed a suggestion of Gödel that the definability function used to build the levels of the constructible hierarchy be modified to make use of stronger logics. The resultant hierarchy might, or might not, then be L itself. We show that by changing the logic in the ramified analytical hierarchy allows one to construct, eg., the minimal ‘correct’ model of analysis.
The next meeting of the Logic Seminar will be in Wednesday, 18/01, 16:00 – 18:00, Ross 70.
Speaker: Chris Lambie-Hanson
Title: Reflections on the coloring and chromatic numbers
Abstract: Compactness phenomena play a central role in modern set theory, and the investigation of compactness and incompactness for the coloring and chromatic numbers of graphs has been a thriving area of research since the mid-20th century,when De Bruijn and Erdős published their compactness theorem for finite chromatic numbers.
In this talk, we will briefly review some of the highlights in this area and then present new results indicating, firstly, that the coloring number can only exhibit a limit amount of incompactness, and, secondly, that large amounts of incompactness for the chromatic number are compatible with strong compactness statements, including compactness for the coloring number.
This is joint work with Assaf Rinot.
Ralf Schindler (Münster) – A Hamel basis for the reals without choice
The Cohen-Halpern-Levy model N has an infinite set of reals without a countable subset. Answering a question of D. Pincus and K. Prikry from 1975, we show that there is a Hamel basis in N. This is joint work with Liuzhen Wu and Liang Yu, inspired by earlier joint work with Mariam Beriashvili. DC fails in N, and it remains open if in the base theory ZF+DC, the existence of a Hamel basis implies that the reals can be wellordered.
This is an announcement of the forthcoming thematic semester on Descriptive Set Theory and Polish Groups that will take place at the Bernoulli Center in Lausanne, Switzerland, from January 1st to June 30th, 2018.
The focus of the semester will be on Descriptive Set Theory and Polish Groups along with applications in other branches of mathematics. While there will be continuous activity at the center throughout the semester, three conferences and two additional workshops will provide the main events. The themes of these will be
Borel combinatorics and ergodic theory
Structure and dynamics of Polish groups
Descriptive set theory
Ideals and exceptional sets in Polish spaces
Large scale geometry of Polish groups
Tomorrow we continue our seminar in Logic, Set Theory and Topology.
Time: Tuesday, January 3, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Salma Kuhlmann (Konstantz)
Title: The Baer-Krull Theorem for Quasi-ordered fields
In my seminar talk on 29.12.2015, I introduced the notion of quasi-ordered fields, proved Fakhruddin’s dichotomy. In this talk, I will present a version of a classical theorem in real algebra (the Baer-Krull theorem) for quasi-ordered fields.
The next meeting of the Logic Seminar will be in Wednesday, 04/01/17, between 16:00 – 18:00, Ross 70.
Please forward this mail to anyone that might be interested.
Speaker: Ludomir Newelski
Title:Stable groups and topological dynamics
Abstract: Assume G is a stable group. I will recall an old 2-step theorem of mine on generating a type-definable subgroup of G by a single type. I will discuss some related questions and put them into context of topological dynamics.
A subset $F$ of a regular uncountable cardinal $\kappa$ is said to be fat iff for every club $C\subseteq\kappa$, and every ordinal $\alpha<\kappa$, $F\cap C$ contains a closed copy of $\alpha+1$.
By a theorem of H. Friedman from 1974, every stationary subset of $\omega_1$ is fat. In particular, $\omega_1$ may be partitioned into $\omega_1$ many pairwise disjoint fat sets.
In this talk, I shall prove that $\square(\kappa)$ give rise to a partition of $\kappa$ into $\kappa$ many pairwise disjoint fat sets. In particular, the following are equiconsistent:
$\omega_2$ cannot be partitioned into $\omega_2$ many pairwise disjoint fat sets;
$\omega_2$ cannot be partitioned into two disjoint fat sets;
The next meeting of the Logic Seminar will be in Wednesday, 28/12/16, 16:00 – 18:00, in Ross Building, 70.
Title: Better lucky than smart: realizing a quasi-generic class of measure preserving transformations as diffeomorphisms. Speaker: Matthew Foreman Abstract: In 1932, von Neumann proposed classifying measure preserving diffeomorphisms up to measure isomorphism. Joint work with B. Weiss shows this is impossible in the sense that the corresponding equivalence relation is not Borel; hence impossible to capture using countable methods.
An accidental consequence of the proof addresses a different classical problem: which measure preserving transformations are isomorphic to diffeomorphisms of a compact smooth manifold?
In this talk we discuss the proof that a quasi-generic class of measure preserving transformations are isomorphic to measure preserving diffeomorphisms of the torus.