Category Archives: Seminars

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.

Ana Njegomir: A forcing characterization of lambda-ineffable cardinals

Monday, May 29, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Ana Njegomir (Universität Bonn)

Title: A forcing characterization of lambda-ineffable cardinals

Katrin Tent: Ample geometries of finite Morley rank

HUJI Logic Seminar

The Logic Seminar will take place in Wednesday, 24/05, 16:00 – 15:30, in Ross 70.
Speaker: Katrin Tent

Title: Ample geometries of finite Morley rank

Abstract: I will explain the model theoretic notion of ampleness and present the geometric context of recent constructions.

Dana Bartosova: When can we act freely?

Place: Fields Institute (Stewart Library)

Date: May 19, 2017 (13:30-15:00)

Speaker: Dana Bartosova, CMU

Title: When can we act freely?

Abstract:  A topological group admits a free action if there is a compact
Hausdorff space on which the group acts without fixed points. I will
discuss this notion and explain how to translate it into colourings of
graphs and Ramsey type properties.

Jarosław Swaczyna: Haar-small sets

Tuesday, May 23, 2017, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Jarosław Swaczyna (Lodz University of Technology)

Title: Haar-small sets

Abstract:

In locally compact Polish groups there is a very natural $\sigma$-ideal of null sets with respect to Haar-measure. In non locally compact groups there is no Haar measure, however Christensen introduced a notion of Haar-null sets which is an analogue of locally compact case. In 2013 Darji introduced a similar notion of Haar-meager sets. During my talk I will present some equivalent definition of Haar-null sets which leads us to joint generalization of those notions. This is joint work with T. Banakh, Sz. Głąb and E. Jabłońska.

Victoria Gitman: A model of second-order arithmetic with the choice scheme in which $\Pi^1_2$-dependent choice fails

KGRC research seminar – 2017‑05‑18 at 4pm

Speaker: Victoria Gitman (CUNY Graduate Center, New York, USA)

Abstract: 

In second-order arithmetic, the choice scheme is the scheme of assertions, for every second-order formula $\varphi(n,X,A)$, that if for every $n$ there is a set $X$ such that $\varphi(n,X,A)$ holds, then there is a single set $Y$ whose $n$-th slice $Y_n$ witnesses $\varphi(n,Y_n,A)$. While full second-order arithmetic ${\textrm Z}_2$ implies the choice scheme for $\Sigma^1_2$-assertions, the reals of the Feferman-Lévy model form a model of ${\textrm Z}_2$ in which $\Pi^1_2$-choice fails. The dependent choice scheme is the analogue ${\textrm DC}$ for second-order arithmetic and it asserts, for every second-order formula $\varphi(X,Y,A)$, that if for every set $X$ there is another set $Y$ such that $\varphi(X,Y,A)$ holds, then there is a single set $Z$, viewed as an $\omega$-sequence of sets, such that for every $n$, $\varphi(Z\upharpoonright n,Z_n,A)$ holds. The theory ${\textrm Z}_2$ implies $\Sigma^1_2$-dependent choice, and Simpson has conjectured that there is a model of ${\textrm Z}_2$ with the choice scheme in which $\Pi^1_2$-dependent choice fails. We prove Simpson’s conjecture by constructing a symmetric submodel of a forcing extension in which ${\textrm AC}_\omega$ holds, but ${\textrm DC}$ fails for a $\Pi^1_2$-definable relation on the reals.

We force over $L$ with a tree iteration of Jensen’s forcing (a ccc subposet
of Sacks forcing adding a unique generic real) along the tree ${}^{\lt\omega}\omega_1$, adding a tree, isomorphic to ${}^{\lt\omega}\omega_1$, of finite sequences of reals ordered by extension, such that that the sequences on level $n$ are $L$-generic for the $n$-length iteration of Jensen’s forcing. We extend the uniqueness of generic reals properties of Jensen’s forcing (obtained earlier by Jensen and later by Lyubetsky and Kanovei) by showing that in the tree iteration extension, the only sequences of reals $L$-generic for the $n$-length iteration of Jensen’s forcing are those explicitly added on level $n$ of the generic tree. The uniqueness property implies that the generic tree is $\Pi^1_2$-definable.

The theorem arose out of our attempts to separate the analogues of the
choice scheme and the dependent choice scheme over Kelley-Morse set theory,
and we conjecture that an appropriate generalization of our arguments will
now achieve this result.

This is joint work with Sy-David Friedman.

Nick Ramsey: Definability in the absolute Galois group of a PAC field

HUJI Logic Seminar

The next meeting of the Logic Seminar will be in 15/05, 14:00 – 16:00 in Sprinzak Building, Room 101.  Please note the change in time and place.
Definability in the absolute Galois group of a PAC field

Speaker: Nick Ramsey
Abstract:  A field K is called pseudo-algebraically closed (PAC), if every absolutely irreducible variety defined over K has a K-rational point.  This class of fields was introduced in 1968 by Ax on the way to his famous algebraic characterization of the pseudo-finite fields: a field is elementarily equivalent to an ultra-product of finite fields if and only if it is perfect, has free profinite absolute Galois group on a single generator, and is PAC.  In the decades that followed, the PAC fields were an intensive object of study within model-theoretic algebra.  Cherlin, van den Dries, and MacIntyre gave complete invariants for the first-order theory of a PAC field in terms of its characteristic, absolute numbers, and the ‘co-theory’ of its absolute Galois group, in a formalism they called ‘co-logic’.  This ‘co-logic’ approach was presented within first-order logic by Chatzidakis who, later on, showed how to connect model-theoretic properties of the theory of the absolute Galois group of a PAC field to the model-theoretic properties of the field itself.  We will survey this work in detail and give some recent applications to classification-theoretic questions concerning PAC fields.  Time-permitting, we will also talk about some work in progress and many open problems in this area.

Stefan Hoffelner: $\text{NS}_{\omega_1}$ saturated and a $\Sigma^{1}_{4}$-definable wellorder on the reals

KGRC Research Seminar – 2017‑05‑11 at 4pm.

Speaker: Stefan Hoffelner (KGRC)

Abstract: The investigation of the saturation of the nonstationary ideal $\text{NS}_{\omega_1}$ has a long tradition in set theory. In the early 1970’s K. Kunen showed that, given a huge cardinal, there is a universe in which $\text{NS}_{\omega_1}$ is $\aleph_2$-saturated. The assumption of a huge cardinal has been improved in the following decades, using very different techniques, by many set theorists until S. Shelah around 1985 realized that already a Woodin cardinal is sufficient for the consistency of the statement “$\text{NS}_{\omega_1}$ is saturated”.

Due to work of H. Woodin on the one hand and G. Hjorth on the other, there is a surprising and deep connection between definable wellorders of the reals and the saturation of $\text{NS}_{\omega_1}$: In a universe with a measurable cardinal and $\text{NS}_{\omega_1}$ saturated, it is impossible to have a $\Sigma^1_3$-wellorder. This leads naturally to the question whether there is a universe in which $\text{NS}_{\omega_1}$ is saturated and its reals have a
$\Sigma^1_{4}$-wellorder. In my talk I will outline a proof that this is indeed the case; assuming the existence of $M_1^{\#}$ there is a model with a $\Sigma^1_{4}$-definable wellorder on the reals in which  $\text{NS}_{\omega_1}$ is saturated.

This is joint work with Sy-David Friedman.