Archives of: Israeli Logic Talks

Eilon Bilinsky: Uncountable set of reals with a single accumulation point

BIU seminar in Set Theory

On 20/11/2017, 13-15, Building 505, Room 65

Speaker: Eilon Bilinsky (TAU)

Title: Uncountable set of reals with a single accumulation point

Abstract. We construct a model of ZF with an uncountable set of reals having a unique condensation point. This answers a question of Sierpinski from 1918.

Chris Lambie-Hanson: Squares, ascent paths, and chain conditions

BIU seminar in Set Theory

On 06/11/2017, 13-15, Building 505, Room 65

Speaker: Chris Lambie-Hanson

Title: Squares, ascent paths, and chain conditions

Abstract. Two topics of interest in modern set theory are the productivity of chain conditions and the existence of higher Aronszajn trees. In this talk, we discuss generalizations of both of these topics and their connections with various square principles. In particular, we will prove that, if $\kappa$ is a regular uncountable cardinal and $\square(\kappa)$ holds, then:
1) for all regular $\lambda < \kappa$, there is a $\kappa$-Aronszajn tree with a $\lambda$-ascent path;
2) there is a $\kappa$-Knaster poset $\mathbb{P}$ such that $\mathbb{P}^{\aleph_0}$ is not $\kappa$-c.c.
Time permitting, we will also present a complete picture of the relationship between the existence of special trees and the existence of Aronszajn trees with ascent paths at the successor of a regular cardinal. This is joint work with Philipp Lücke.

Yair Hayut: Magidor cardinal and Magidor filters

BIU seminar in Set Theory

On 30/10/2017, 13-15, Building 505, Room 65

Speaker: Yair Hayut (TAU)

Title: Magidor cardinal and Magidor filters

Abstract. In this talk I will define the notion of Magidor Cardinal (\omega bounded Jonsson cardinal) which is a generalization of Jonsson cardinal.
I will show that the analog of Jonsson filter for Magidor cardinals
is inconsistent with ZFC.

This lecture is based on a joint work with Shimon Garti and Saharon Shelah.

Shimon Garti: The club principle and the splitting number

HUJI Logic Seminar
The next meeting of the Logic Seminar will take place Wednesday, 28/06, 16:00 – 18:00, Ross 70.  This is the last meeting of the Logic Seminar for this semester.
Speaker: Shimon Garti
Title: Tiltan
Abstract: We shall try to prove some surprising (and hopefully, correct) theorems about the relationship between the club principle (Hebrew: tiltan) and the splitting number, with respect to the classical s at omega and the generalized s at supercompact cardinals.

Bill Chen: Variations of the stick principle

BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, June 27th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Bill Chen (BGU)

Title: Variations of the stick principle

Abstract:
The stick principle asserts that there is a family of infinite subsets of $\omega_1$ of size $\aleph_1$ so that any uncountable subset of $\omega_1$ has some member of the family as a subset. We will give a forcing construction to separate versions of the stick principle which put a bound on the order-type of the subsets in the family. Time permitting, we will say a little about the relation of the stick principle with the existence of Suslin trees.

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.​

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.

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.

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.

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.