Archives of: Israeli Logic Talks

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.

Grigory Mashevitzky: Action of endomorphism semigroups on definable sets

​​​​BGU seminar in Logic, Set Theory and Topology

Time: Tuesday, May 9th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Grigory Mashevitzky (BGU)

Title: Action of endomorphism semigroups on definable sets

Abstract:
I plan to discuss the construction, examples and some applications the Galois-type correspondence between subsemigroups of the endomorphism semigroup End(A) of an algebra A and sets of formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of End(A) on definable sets over A. Between possible applications of this Galois correspondence is a uniform approach to geometries defined by various fragments of the initial language.

The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way one can get an effective syntactical expression by semantic tools.

Yet another advantage is a common approach to generalizations  of the main model theoretic concepts to the sublanguages of the first order language. It also reveals new connections between well-known concepts. One more application concerns the generalization of the unification theory or more generally Term Rewriting Theory to the logic unification theory.

Asaf Karagila: The Bristol model: A few steps into a Cohen real

HUJI Logic Seminar

The next meeting will be at 08/05 (in one week), Ross 63, 12:00-14:00.

Title: The Bristol model: A few steps into a Cohen real
Speaker:
Asaf Karagila
Abstract: We will take a close look at the first few steps of the construction of the Bristol model, which is a model intermediate to L[c], for a Cohen real c, satisfying V\neq L(x) for all x.

Ashutosh Kumar: On possible restrictions of null and meager ideal

BIU seminar in Set Theory

On 04/05/2017, 10-12, Building 604, Room 103

Speaker:  Ashutosh Kumar

Title: On possible restrictions of null and meager ideal

Abstract. Fremlin asked if the null ideal restricted to a non null set of reals could be isomorphic to the non stationary ideal on omega_1. Eskew asked if the null and the meager ideal could both be somewhere countably saturated. We’ll show that the answer to both questions is yes. Joint work with Shelah.

Nicholas Ramsey: NSOP_1 Theories

BGU seminar in Logic, Set Theory and Topology.

Time: Tuesday, April 25th, 12:15-13:30.
Place: Seminar room -101, Math building 58.
Speaker: Nicholas Ramsey (UC Berkeley)

Title: NSOP_1 Theories

Abstract:
The class of NSOP_1 theories was isolated by Džamonja and Shelah in the mid-90s and later investigated by Shelah and Usvyatsov, but the theorems about this class were mainly restricted to its syntactic properties and the model-theoretic general consensus was that the property SOP_1 was more of an unimportant curiosity than a meaningful dividing line. I’ll describe recent work with Itay Kaplan which upends this view, characterizing NSOP_1 theories in terms of an independence relation called Kim-independence, which generalizes non-forking independence in simple theories.  I’ll describe the basic theory and describe several examples of non-simple NSOP_1 theories, such as Frobenius fields and vector spaces with a generic bilinear form.

Thilo Weinert: Partitioning pairs of sigma-scattered linear orders

BIU seminar in Set Theory

On 20/04/2017, 10-12, Building 604, Room 103

Speaker:  Thilo Weinert

Title: Partitioning pairs of sigma-scattered linear orders

Abstract. We are going to continue the analysis of generalised scattered orders, proving the result described towards the end of Chris Lambie-Hanson’s talk. This states that consistently, for every sigma-scattered linear ordering there is a colouring of its pairs in black & white such that every triple contains a white pair and every copy of one of the following order-types contains a black pair:

  • omega_1^omega
  • (omega_1^omega)^*
  • omega_1 * (omega * omega^*)^omega
  • omega_1^* * (omega * omega^*)^omega
  • (omega * omega^*)^omega * omega_1
  • (omega * omega^*)^omega * omega_1^*

This generalises a 46-year-old Theorem of Erdős & Rado about ordinals. A sufficient hypothesis implying this theorem is the existence of a colouring of the pairs of omega_1 * omega in black & white such that every triple contains a black pair and every subset of full order-type contains a white one. Time permitting we may present a proof that stick = b = Aleph_1 implies the existence of such a colouring. Here b is the unbounding number and stick = Aleph_1 is a weakening of the club principle which was considered by Baumgartner 41 years ago, named by Broverman, Ginsburg, Kunen & Tall two years thereafter and twenty years ago reconsidered as a cardinal characteristic by Fuchino, Shelah & Soukup.