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.
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.
An infinite series of real numbers is conditionally convergent if it converges, but the sums of the positive and of the negative terms are both divergent. How many infinite subsets of the naturals are necessary such that every conditionally convergent series has a subseries given by one of our infinite subsets that is divergent? The answer to this question is known as the subseries number ß, and was isolated as a cardinal characteristic of the continuum by Brendle, Brian and Hamkins.
The intermediate model theorem states that whenever G is generic over V and V ⊆ M ⊆ V[G] are models of ZFC, then M is also a forcing extension of V . Unfortunately, this fails completely if we only assume ZF instead. Can more can be said? The goal of our talk is to present a generalization of the above theorem that works for ZF and talk about some of the recent progress made in the theory of symmetric extensions. This is joint with A. Karagila.
Martin's Conjecture is a proposed classification of Turing-invariant functions under the Axiom of Determinacy. Whether the classification holds for the ostensibly smaller class of order-preserving functions is open, but more tractable. In this talk, we’ll explain an approach to proving Martin’s Conjecture for order-preserving functions and discuss how far we can go. This is joint work with Patrick Lutz.
Henselian valued fields are a class of structures whose model theory has been much investigated. After an introduction to this area, I will present recent joint work with Anscombe and Jahnke in the finitely ramified setting. Prior knowledge of valued fields will not be assumed.
In this talk, we consider how much non-constructive principles are sufficient for Friedberg-Muchinik construction of degree $d$ such that $0<d<0'$. We will see that the only point we need a non-constructive principle is to show "if a recursive set $S$ of natural number has finite cardinality, then $S$ has an upper bound", which requires $\Sigma^0_1$ law of excluded middle.
Let $\mathsf{M}$ be the weak set theory (with powersets) axiomatised by: $\textsf{Extensionality}$, $\textsf{Pair}$, $\textsf{Union}$, $\textsf{Infinity}$, $\textsf{Powerset}$, transitive containment ($\textsf{TCo}$), $\Delta_0\textsf{-Separation}$ and $\textsf{Set-Foundation}$. In this talk I will discuss the relationship between two alternative versions of the set-theoretic collection scheme: $\textsf{Collection}$ and $\textsf{Strong Collection}$. Both of these schemes yield $\mathsf{ZF}$ when added to $\mathsf{M}$, but when restricted the $\Pi_n$-formulae (denoted $\Pi_n\textsf{-Collection}$ and $\textsf{Strong } \Pi_n\textsf{-Collection}$) these alternative versions of set-theoretic collection differ. In particular, over the theory $\mathsf{M}$, $\textsf{Strong }\Pi_n\textsf{-Collecton}$ is equivalent to $\Pi_n\textsf{-Collection}+\Sigma_{n+1}\textsf{-Separation}$. And, $\mathsf{M}+\textsf{Strong }\Pi_n\textsf{-Collection}$ proves the consistency of $\mathsf{M}+\Pi_n\textsf{-Collection}$. In this talk I will show that, despite this difference in consistency strength, every countable well-founded model of $\mathsf{M}+\Pi_n\textsf{-Collection}$ satisfies $\textsf{Strong } \Pi_n\textsf{-Collection}$. If time permits I will outline how this argument can be refined to show that $\mathsf{M}+\Pi_n\textsf{-Collection}+\Pi_{n+1}\textsf{-Foundation}$ proves $\Sigma_{n+1}\textsf{-Separation}$.
We study the structure of the convolution semigroups of measures over definable groups. We isolate the property of generic transitivity and demonstrate that it is sufficient (and necessary) to develop stable group theory localizing on a generically stable type. We establish generic transitivity of generically stable idempotent types in important new cases, including abelian groups in arbitrary theories and arbitrary groups in rosy theories, and characterize them as generics of connected type-definable subgroups. Using tools from Keisler's randomization theory, we generalize some of these results from types to generically stable Keisler measures, and classify idempotent generically stable measures in abelian groups as (unique) translation-invariant measures on type-definable fsg subgroups. This provides a partial definable counterpart to the classical work of Rudin, Cohen and Pym for locally compact topological groups. Working over a countable NIP structure, we provide an explicit construction of a minimal left ideal in the convolution semigroup of measures from a minimal left ideal of types and the unique Haar measure on the ideal group. As a key ingredient, we prove the revised Ellis group conjecture of Newelski saying that under NIP, the so-called tau-topology on the ideal group is Hausdorff.
Joint work with Kyle Gannon and Krzysztof Krupiński.
