Ari Brodsky: ​Constructing free Souslin trees from a proxy principle

BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, June 13th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: ​Ari Brodsky (BIU)

Title: ​Constructing free Souslin trees from a proxy principle

Abstract. ​More than 40 years ago, Jensen constructed a free Souslin tree of height $\omega_1$ from $\diamondsuit(\omega_1)$.
We show how to construct a free $\kappa$-Souslin tree, where $\kappa$ is an arbitrary regular uncountable cardinal.
This is joint work with Assaf Rinot.​

Aleksander Cieślak: Cohen-stable families of subsets of integers

Tuesday, June 13, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Aleksander Cieślak (Wroclaw University of Science and Technology)

Title: Cohen-stable families of subsets of integers

Abstract:

A mad family is Cohen-stable if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-stable. We will find condition necessary and sufficient for mad family to be Cohen-unstabe and investigate when such family exist.

Yizheng Zhu: Iterates of M_1

Monday, June 12, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Yizheng Zhu (University of Münster)

Title: Iterates of M_1

Abstract:

Assume Delta^1_3-determinacy. Let L_{kappa_3}[T_2] be the admissible closure of the Martin-Solovay tree and let M_{1,infty} be the direct limit of$M_1 via countable trees. We show that L_{kappa_3}[T_2]cap V_{u_{omega}} = M_{1,infty} | u_{omega}.

Thilo Weinert: Avoiding Quadruples Using a Scale

BIU seminar in Set Theory

On 08/06/2017, 10-12, Building 604, Room 103

Speaker: Thilo Weinert (BGU)

Title: Avoiding Quadruples Using a Scale

Abstract. In 1971, Hajnal showed that the continuum hypothesis implies omega_1^2 -|-> (omega_1^2, 3)^2 and in the same year, together with Erdős, that GCH implies that for every infinite cardinal kappa and every alpha < (kappa^+)^2 we have alpha -|-> (kappa^+ * kappa, 3)^2. In the same paper they showed that for infinite cardinals kappa and alpha < (kappa^+)^2 we have (kappa^+)^2 —> (alpha, 3)^2. In 1987, together with Baumgartner, he showed that for regular kappa satisfying 2^kappa = kappa^+ = lambda we have lambda^2 -|-> (lambda * kappa, 4)^2.

In 1998, Jean Larson showed that for regular kappa and lambda = kappa^+ the existence of a scale of length lambda of functions f : kappa — > kappa implies the failure of the aforementioned partition relations shown to
be unprovable from ZFC in the seventies, i.e. lambda * kappa —> (lambda * kappa, 3)^2 and lambda^2 —> (lambda^2, 3)^2. She commented that it would be interesting to know whether this hypothesis also suffices to prove lambda^2 -|-> (lambda * kappa, 4)^2.

It does.

Zoltán Vidnyánszky: Borel chromatic numbers: finite vs infinite

KGRC Research Seminar – 2017-06-08 at 4pm.

Speaker: Zoltán Vidnyánszky (York University, Toronto, Canada)

Abstract: One of the most interesting results of Borel graph combinatorics is the $G_0$ dichotomy, i. e., the fact that a Borel graph has uncountable Borel chromatic number if and only if it contains a Borel homomorphic image of a graph called $G_0$. It was conjectured that an analogous statement could be true for graphs with infinite Borel chromatic number. Using descriptive set theoretic methods we answer this question and a couple of similar questions negatively, showing that one cannot hope for the existence of a Borel graph whose embeddability would characterize Borel (or even closed) graphs with infinite Borel chromatic number. We will also discuss a positive result and its relation to Hedetniemi’s conjecture.

Assaf Hasson: Strongly dependent henselian fields and ordered abelian groups

​​BGU Seminar in Logic, Set Theory and Topology

Time: Tuesday, June 6th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Assaf Hasson (BGU)

Title: Strongly dependent henselian fields and ordered abelian groups

Abstract:
The strong non-independence property was introduced by Shelah in order to capture, within the class of theories without the independence property (aka dependent theories), an analogue of the class of super-stable theories. Shelah conjectured (roughly) that any infinite field with the strong non-independence property (aka strongly dependent) is either real closed, algebraically closed or supports a definable (henselian) valuation. The conjecture was solved (Johnson) in the very special case of dp-minimal fields, and otherwise remains wide open. In fact, most experts believe the conjecture (replacing “algebraically closed” with “separably closed”) to be true of all fields without the independence property, and the algebraic division line between the two classes of fields remains unclear.

In the talk we will show that strongly dependent ordered abelian groups do have a simple algebraic characterisation, and suggest the interpretability of ordered  abelian groups which are not strongly dependent as a new (not yet fully satisfactory) conjectural division line.

If time allows we will draw from the classification of strongly dependent ordered abelian groups some conclusions concerning strongly dependent henselian fields (e.g., if K is strongly dependent then any henselian valuation v — not necessarily definable — on K has strongly dependent residue field and value group).

The talk will aim to be, more or less, self-contained and little use (if any) will be made of technical model theoretic terms.

Based (mostly) on joint work with Yatir Halevi.

Natasha Dobrinen: The big Ramsey degrees for the universal triangle-free graph

Mathematical logic seminar – Jun 5 2017
Time:     3:00pm – 4:00 pm

Room:     Wean Hall 8220

Speaker:         Natasha Dobrinen
Department of Mathematics
University of Denver

Title:     The big Ramsey degrees for the universal triangle-free graph

Abstract:

The Rado graph (aka the countable random graph) is the unique countable graph G which is:

a) Universal, that is G contains an induced copy of every finite graph.

b) Homogeneous, that is any isomorphism between finite induced subgraphs
of G extends to an automorphism of G.

The construction of the Rado graph works with many classes of finite structures (the Fraïssé classes), assigning to each Fraïssé class a countable, universal and homogeneous structure called the Fraïssé limit.

Ramsey theory on relational structures can be studied from two vantage points. The first, more classical, is to study when, given two finite structures A and B and given any k greater than 1, there is another finite structure C such that for any coloring of all copies of A in C into k colors, there is a copy of B in C in which all copies of A have the same color. A Fraïssé class of finite relational structures has the Ramsey property if this holds for any two structures A and B in the class. Nešetřil and Rödl have shown that many classes of finite ordered relational structures have the Ramsey property, including finite ordered graphs and finite ordered triangle-free graphs.

The second, and of much recent interest, is to study colorings of copies of a finite structure inside an infinite homogenous structure, usually the Fraïssé limit of some Fraïssé class of finite structures. It has been shown that any finite coloring of the vertices of the Rado graph can be reduced to one color on a subgraph which is also a Rado graph. For edges and other structures with more than one vertex, Sauer has proved this to be impossible. However, he also proved that given a finite graph A, there is a number n(A) such that any coloring of all copies of A in the Rado graph into finitely many colors may be reduced to n(A) colors on a copy of the Rado graph. We say, then, that the Rado graph has finite big Ramsey degrees. Similar results have been obtained for other countable homogeneous structures, though many are still open.

We have looked at the problem of finite big Ramsey degrees for the universal triangle-free graph H, that is, the homogeneous graph with no triangles into which every countable triangle-free graph embeds. This is the first homogeneous structure omitting a subtype to be addressed for big Ramsey degrees. Using the method of forcing, but in ZFC, we prove a new Ramsey theorem on trees which code H, and apply it to deduce that H has finite big Ramsey degrees.

Chris Scambler: On Ineffable Liars

KGRC Friday seminar on 2017‑06‑02 – 12 pm

Speaker: Chris Scambler (New York University, USA)

Abstract: The most promising non-classical approaches to the theory of truth build on that of Saul Kripke (1975) by adding a conditional satisfying reasonable laws. Among the attractive features of such approaches are their capacity to offer object-language means for classifying the defectiveness of paradoxical sentences and formulas; in (2007), Hartry Field shows his approach yields a transfinite hierarchy of determinacy operators of increasing strength that seem to play exactly this role. There are, however, difficult technical questions about the extent of the hierarchy of such operators that turn on the availability of reasonable ordinal notation systems, and these may yield philosophical issues for Field’s approach to the paradoxes. According to Field, the extent of the hierarchy is inherently ‘fuzzy’, because of indeterminacy concerning the unrestricted notion of definability. As a result, Field argues, one can’t diagonalize out of the hierarchy of determinacy operators in any meaningful sense, since the hierarchy in question is not bivalently definable. In (2014), Philip Welch has argued that on the contrary the hierarchy of determinacy operators breaks down precisely at the least $\Sigma_2$-extendible ordinal (relative to a given model M); moreover, Welch has shown how to use this result to produce “ineffable liars”, that diagonalize out of the hierarchy: these are sentences that are indeterminate on Field’s theory, but whose defectiveness is not measured by any determinacy operator in the object language.

The task of this paper is to assess the significance of Welch’s result, and to adjudicate the dispute between Field and Welch. In the opening sections, I will review the Kripke and Field constructions, focussing especially on the hierarchy of determinacy operators and their behaviour. After that, I will give an overview of Welch’s construction, culminating in the construction of an ineffable liar sentence. Finally, I will scrutinize Welch’s argument from a philosophical perspective, and suggest that Field’s project is not adversely affected by Welch’s results. Nevertheless, I will show some ways in which that the latter are still of considerable philosophical interest.

References

Saul A. Kripke: Outline of a theory of truth. Journal of Philosophy 72 (19):690-716 (1975)

Hartry Field: Solving the paradoxes, escaping revenge. In J. C. Beall (ed.), Revenge of the Liar: New Essays on the Paradox. Oxford University Press (2007)

P. D. Welch: Some observations on truth hierarchies. Review of Symbolic Logic 7 (1):1-30 (2014)

 

Vincenzo Dimonte: Rank-into-rank axioms and forcing

KGRC Research Seminar – 2017‑06‑01 at 4pm

Speaker: Vincenzo Dimonte (University of Udine, Italy)

Abstract: Rank-into-rank axioms sit on the top of the large cardinal hierarchy, and their fringe status makes them quite mysterious and evasive. In particular, research on I0 started to gain momentum just in the last few years.

In this talk we will give an overview of what is known at the moment about the interaction between such axioms and forcing, in four steps, in increasing order of complexity. The main result most of the time would be that the rank-into-rank axiom is not destroyed by the forcing, therefore providing many independence results (for example involving the behaviour of the power function, tree structures, pcf theory…). We will also note how such results pose an actual problem for the main branch of the research on I0, i.e, the quest for finding similarities between I0 and the Axiom of Determinacy.

Part of this work is joint with Sy Friedman and Liuzhen Wu.

Symposium of the Set Theoretic Pluralism network, Bristol, June 20-25

The second and final symposium of the Set Theoretic Pluralism (STP) network will take place at the University of Bristol on June 20-25, 2017. We would like to invite researchers from all relevant disciplines to attend the symposium, including set theory, philosophy of mathematics, metaphysics, philosophy of language, and epistemology.

The STP network draws together experts in mathematics and philosophy to grapple with the increasingly popular idea that mathematical reality may be best understood as fractured and indeterminate. Participants will include Sy David Friedman (Vienna), Peter Koellner (Harvard), Hugh Woodin (Harvard), Sean Walsh (UC Irvine), Toby Meadows (Queensland), Philip Welch (Bristol), Giorgio Venturi (Campinas), Carolin Antos (Vienna), Neil Barton (Vienna), Zeynep Soysal (Harvard), and Benedict Eastaugh (Bristol).

Registration for the symposium is now open (participation is free, but registration is required). The registration deadline is Monday 5 June, 2017.

To register, please email benedict.eastaugh@bristol.ac.uk and include the following information:

* Name and affiliation.

* Whether you would like to attend the conference dinner, scheduled for Thursday 22 June, 2017. Places are limited and will be allocated on a first-come, first-served basis.

* Any dietary restrictions or preferences.

* Which days you plan to attend the symposium, if you are not attending the entire event.

Further details of the symposium, including the programme and details of the location, will be published on the symposium website:

https://sites.google.com/site/pluralset/symposium-2

The STP network is funded by a Leverhulme International Network grant and is hosted at the University of Bristol. The STP network partners include the University of Aberdeen, Harvard University, the University of Bristol, the University of Vienna, the University of Helsinki, and the University of California, Irvine. For more details of the network’s context and aims, please refer to the network website:

https://sites.google.com/site/pluralset/context-aims-of-the-network

We look forward to seeing you here!

Toby Meadows, Philip Welch, and Benedict Eastaugh