The union of all the seminar of the Logic group: Logic (Wed 4pm), Model Theory (Wed 2pm), Set Theory (Wed 1pm).
Results 21 to 30 of 41
An exact linear order is one with no non-trivial self-embedding. I shall talk a little bit about these objects and some questions of interest surrounding them, taking us on a path through Ramsey constructions and curious questions in abstract forcing.
The notion of primitive permutation group for group actions resembles that of simple group for abstract groups. I will discuss work in progress with Katrin Tent (building on earlier work with Liebeck and Tent) on primitive pseudofinite permutation groups. Some methods are analogous to those used in John Wilson’s classification of simple pseudofinite groups.
Dependent Choice (DC) is one the most useful choice principles with many equivalents (including the Downward Löwenheim–Skolem and the Baire Category Theorem). When we violate the Axiom of Choice via symmetric extensions we often want to preserve at least that much. In this talk we will discuss a few older results about the preservation of DC in generic and symmetric extensions, and we will present a recent breakthrough from a work-in-progress with Jonathan Schilhan.
Dependent Choice (DC) is one the most useful choice principles with many equivalents (including the Downward Löwenheim–Skolem and the Baire Category Theorem). When we violate the Axiom of Choice via symmetric extensions we often want to preserve at least that much. In this talk we will discuss a few older results about the preservation of DC in generic and symmetric extensions, and we will present a recent breakthrough from a work-in-progress with Jonathan Schilhan.
The open core of a structure is the reduct generated by the open definable sets. Tame topological structures (e.g. o-minimal) are inter-definable with their open core. Structures such as M = (ℝ, <, +, ℚ) are wild in the sense that they define a dense co-dense set. Still, M is NIP and its open core is o-minimal. In this talk we push forward the thesis that the open core of an NTP2 (a generalization of NIP) topological structure is tame. Our main result is that, under suitable conditions, the open core has quantifier elimination, and its definable functions are generically continuous.
Fundamental to the practice of logic is the dogma regarding the first order/second order logic distinction, namely that it is ironclad. Was it always so? The emergence of the set theoretic paradigm is an interesting test case. Early workers in foundations generally used higher order systems in the form of type theory; but then higher order systems were gradually abandoned in favor of first order set theory—a transition that was completed, more or less, by the 1930s.
As for logic in general, the concept of a logic being first order is not only about whether the variables range over the elements of a given domain, or over sets of elements, or over sets of sets of elements, and so on; it is also, I suggest, about the context.
Of course, set theory is a theory and second order logic is a logic, at least that is the common understanding. However if one cares to view set theory as a logic—and if we do think of set theory as a logic, it is a logic with the cumulative hierarchy 𝑉 as its standard (class) model—then set theory turns out to be a stronger logic than second order logic. This is perhaps as it should be, given that the latter restricts the domain of quantifiable objects to those generated by (at most) a single iteration of the power set operation, while set theory allows for arbitrary iterations of the power set operation.
This talk is based on the forthcoming paper "How first order is first order logic?" by J. Kennedy and Jouko Väänänen for The Oxford Handbook of Philosophy of Logic. Editors: Elke Brendel, Massimiliano Carrara, Filippo Ferrari, Ole Hjortland, Gil Sagi, Gila Sher, Florian Steinberger, Oxford University Press.
One of the strongest known large cardinal axioms is I3, positing the existence of a non-trivial elementary embedding $j$ from $V_λ$ to $V_λ$ for some $λ$. Given two such embeddings $j$ and $k$ for the same lambda, there is a natural "application" operation to yield a third, $j*k$, and elementarity shows that this operation is left self-distributive: $j*(k*l)=(j*k)*(j*l)$. Structures with such an operation are called LD-algebras or shelves. Laver showed that the algebra of embeddings generated by a single such $j$ under $*$ is in fact the free LD-algebra on 1 generator; and the set-theoretic context around this concrete (once you've assumed I3) instantiation of the free LD-algebra gives rise to various theorems about LD-algebras that are only known under this very strong large cardinal assumption. Given I3, there will be many other embeddings from $V_λ$ to $V_λ$, and it is natural to ask if one can obtain from amongst them a free LD-algebra on more than one generator. In joint work with Scott Cramer and Sheila Miller, we show that the answer is positive if one assumes a little more: from I2 we get a free 2-generator LD-algebra of embeddings. This talk will focus on set-theoretic aspects of the proof; a week later I will be giving a talk in the ARTIN conference on the same topic, focusing more on the algebraic aspects.
An element $x$ in a group $G$ is a left Engel element if for each $x ∈ G$ there exists a positive integer $n = n(x)$ such that $[[[g,x],x],··· ,x] = 1$ ($n$ times). If $n = n(x)$ can be chosen independently of $x$, then we say that $x$ is a left $n$-Engel element. There are some connections to groups of prime power exponent and for example, every element in a group of exponent 3 is a left 2-Engel element. Whereas it is easy to see that the normal closure of a left 2-Engel element is abelian, it is still an open question whether the normal closure of a left 3-Engel element is locally nilpotent. We will give some overview of this problem, focusing on advances in recent years.
A valued difference field is a valued field equipped with an automorphism which fixes the valuation ring setwise. I will discuss various properties of the existentially closed valued difference fields, both from the algebraic and the model-theoretic perspective, and I will highlight the role of tropical geometry in some of the proofs. This is joint work with Jan Dobrowolski and Rosario Mennuni.
The problem of integration in finite terms is the problem of finding exact closed forms for antiderivatives of functions, within a given class of functions. Liouville introduced his elementary functions (built from polynomials, exponentials, logarithms and trigonometric functions) and gave a solution to the problem for that class, nearly 200 years ago. The same problem was shown to be decidable and an algorithm given by Risch in 1969.
We introduce the class of exponentially algebraic functions, generalising the elementary functions and much more robust than them, and give characterisations of them both in terms of o-minimal local definability and in terms of definability in a reduct of the theory of differentially closed fields.
We then prove the analogue of Liouville's theorem for these exponentially-algebraic functions.
This is joint work with Rémi Jaoui.