Leeds Logic Day, 25/03/15

Photos

Below are photos of the speakers that were taken during the day:

Schedule

All talks will be held in the Roger Stevens Building, Lecture Theatre 6.

11:10: Coffee/Tea at Dolce Vita Cafe
11:30: Erick Garcia Ramirez, Stratifications of Definable Subsets in Valued Fields
12:15: Ricardo Bello Aguirre, Generalised stability
13:00: Lunch
14:00: Daoud Siniora, Ample Generics
14:45: Anton Freund, Minimal proof sizes via reflection
15:30: Break
15:45: Anja Komatar, Rainbows
16:30: Thomas Forster, The Consistency of NFU
18:00: Dinner

Abstracts

Erick Garcia Ramirez, Stratifications of Definable Subsets in Valued Fields
Historically, stratifications of sets are a useful tool in Algebraic Geometry, particularly in the problem of studying singularities. I will introduce a recent notion of stratifications for definable sets in Henselian valued fields due to I. Halupczok. Then I will describe my work on stratifications of the tangent cone of definable subsets of the reals.

Ricardo Bello Aguirre, Generalised stability
In this talk we will begin talking about stable theories and its relation to the classification problem. We will later mention ideas around simple theories, dependent theories and theories without the tree property of the second kind. We'll also present some examples of each of these classes of theories.

Daoud Siniora, Ample Generics
The automorphism group of a first order structure can be equipped with a topology making it a topological group. Thus, we can talk about the generic automorphisms of the structure, and more generally about ample generics. For the existence of ample generics, it is sufficient for the age of the structure to have a certain amalgamation property and an extension property for partial automorphisms. I will introduce ample generics and talk about the aforementioned sufficient conditions.

Anton Freund, Minimal proof sizes via reflection
The reflection principle for a theory T says that any formula provable in T is true. By Gödel's second incompleteness theorem this statement isn't provable in T itself. More precisely, the reflection principle for an existential formula says that "for every p there is a y such that if p is a proof of the existential formula then y is a witness for which the formula holds". When we read this computationally then the reflection principle postulates a function which maps a proof of an existential formula to a witness for that formula. To quantify the strength of reflection principles (by determining their provably total functions) thus means to give upper bounds for witnesses of the formula in terms of the size of a minimal proof. By contraposition this gives a lower bound for the size of proofs in terms of a minimal witness to the formula. Thus if the formula has only very large witnesses then there cannot be any short proofs of it. As an application of this idea we consider the strengthened finite Ramsey theorem. By the famous result of Paris and Harrington this theorem isn't provable in Peano arithmetic. At the same time any particular instance of the theorem is provable in very weak theories (e.g. Robinson arithmetic). We will see that the proofs of these instances are enormously long in any fragment of Peano arithmetic while they are reasonably short in Peano arithmetic itself.

Anja Komatar, Rainbows**
In this talk, we aim to provide a gentle introduction to Structural Ramsey Theory. Ramsey's Theorem (see previous talk) can be re-phrased as saying that the class of all finite complete graphs is a Ramsey class. Structural Ramsey Theory asks whether other classes of finite structures are also Ramsey.

The focus of the talk will be on providing examples and non-examples, so that by the end of the talk you will have gained some familiarity with the notion of a Ramsey structure.

**The speaker had told me that they would like the word `rainbows' to be in the title. So, I figured, why not?

Thomas Forster, The Consistency of NFU
In 1967 Ronald Jensen (my Doktorgrossvater) proved the consistency of Quine's set theory NF modified to allow atoms. He is quoted as saying that of all his work that was the result that pleased him best. (Mind you, he said that before he did Fine Structure of L!). The result combines Mostowski's technique of Extracted Models, Specker's Equiconsistency of Type-theory-with-ambiguity and NF, Ramsey's theorem, and ultraproducts; and withal it is - altho' unexpected - not at all hard. All in all, a crowd-pleaser.