Models and Sets Seminar

Location: MALL
Title: Introducing a first-order theory of ordered transexponential fields
(joint work with Salma Kuhlmann)
Studying the growth properties of definable functions in o-minimal settings, Miller established the following remarkable growth dichotomy: an o-minimal expansion of an ordered field is either power bounded or admits a definable exponential function (see [2]). Going one step further in the hierarchy of growth, Miller’s dichotomy result naturally led to the question whether there exist o-minimal expansions of ordered fields that are not exponentially bounded. Recent research activity in this area is therefore motivated by the search for either an o-minimal expansion of an ordered exponential field by a transexponential function that eventually exceeds any iterate of the exponential or, contrarily, for a proof that any o-minimal expansion of an ordered field is already exponentially bounded.

In this talk, I will present our axiomatic approach towards the study of ordered fields equipped with a transexponential function from [1]. Namely, denoting by e an exponential and by T a compatible transexponential, we establish a first-order theory of ordered transexponential fields in which the functional equation $T(x + 1) = e(T(x))$ holds for any positive $x$. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural (i.e. the finest non-trivial convex) valuation. Moreover, I will illustrate construction methods for transexponentials on non-archimedean ordered exponential fields.
All relevant valuation theoretic background will be introduced.
[1] L. S. Krapp and S. Kuhlmann, ‘Ordered transexponential fields’, preprint, 2023, arXiv:2305.04607v1.

[2] C. Miller, ‘A Growth Dichotomy for O-minimal Expansions of Ordered Fields’, Logic: from Foundations to Applications (eds W. Hodges, M. Hyland, C. Steinhorn and J. Truss; Oxford Sci. Publ., Oxford Univ. Press, New York, 1996) 385–399.

Location: MALL
Title: Very large set axioms over Constructive Set Theories
One of the main areas of research in set theory is the study of large cardinal axioms and many of these can be characterised by the existence of elementary embeddings with certain properties. The guiding principle is then that the closer the domain and co-domain of the embedding is to the universe, the stronger the resulting large cardinal axiom. This leads naturally to the question of whether there is an elementary embedding of the universe into itself which is not the identity, and the least ordinal moved by such an embedding is known as a Reinhardt cardinal. While Kunen famously proved that no such embedding can exist if the universe satisfies ZFC, it is an open question in many subtheories of ZFC, most notably ZF (without Choice).

In this talk we will study elementary embeddings in the weaker context of intuitionistic set theories, that is set theories without the law of excluded middle. We shall observe that the ordinals can be very ill-behaved in this setting and therefore we will reformulate large cardinals by instead looking for large sets which capture the desired structural properties. We shall investigate the consistency strength of analogues to measurable cardinals, Reinhardt cardinals and many other similar ideas in terms of the standard ZFC large cardinal hierarchy.

This is joint work with Hanul Jeon.

Location: MALL
Title: A surjection from square onto power
Cantor proves that for any set $A$, there is no surjection from $A$ into its power set $\mathcal{P}(A)$. In this talk, we describe a construction of a ZF model. In this model, there is a set $A$ and a surjection from its square set $A^2$ onto its power set $\mathcal{P}(A)$. This indicates Cantor's Theorem is in some sense optimal. This is joint work with Guozhen Shen and Yinhe Peng.

Location: MALL
Title: Regular solutions of systems of transexponential-polynomial equations

It is unknown whether there are o-minimal fields that are transexponential, i.e., that define functions which eventually grow faster than any tower of exponential functions. In past work, I constructed a Hardy field closed under a transexponential function $E$ which satisfies $E(x+1) = \exp E(x)$. Since the germs at infinity of unary functions definable in an o-minimal structure form a Hardy field, this can be seen as evidence that the real field expanded by $E$ could be o-minimal. To prove o-minimality, a better understanding of definable functions in several variable is likely needed. I will discuss one approach using a criterion for o-minimality due to Lion. This ongoing work is joint with Vincent Bagayoko and Elliot Kaplan.

Location: Roger Stevens LT 16
Title: Building generalized indiscernibles in nonelementary classes
NOTE location change: Roger Stevens LT 16

Generalized indiscernibles can be built in first-order theories by generalizing the combinatorial Ramsey’s Theorem to classes with more structure, which is an active area of study. Trying to do the same for infinitely theories (in the guise of Abstract Elementary Classes) requires generalizing the Erdos-Rado Theorem instead. We discuss various results about generalizations of the Erdos-Rado Theorem and techniques (including large cardinals and forcing) to build generalized indiscernibles.

Location: MALL
Title: Pseudo-finite semigroups and diameter

A semigroup $S$ is said to be (right) pseudo-finite if the universal right congruence $S \times S$ can be generated by a finite set $U$ of pairs of elements of $S$ and there is a bound on the length of derivations for an arbitrary pair as a consequence of those in $U$ . The diameter of a pseudo-finite semigroup is the smallest such bound taken over all finite generating sets.

The notion of being pseudo-finite was introduced by White in the language of ancestry, motivated by a conjecture of Dales and Zelazko for Banach algebras. The property also arises from several other sources.

Without assuming any prior knowledge, this talk investigates the somewhat unpredictable notion of pseudo-finiteness. Some well-known uncountable semigroups have diameter $1$; on the other hand, a pseudo-finite group is forced to be finite. Actions, presentations, Rees matrix constructions and some good old-fashioned semigroup tools all play a part.

This research sits in the wider framework of a study of finitary conditions for semigroups.

Location: MALL
Title: Combining Manin-Mumford and weak Zilber-Pink

I will introduce some classical notions and problems in Diophantine geometry, including the Manin-Mumford and Zilber-Pink conjectures, and explain how model-theoretic tools are used to approach them. I will then talk about one of my recent theorems establishing a new partial result towards Zilber-Pink by combining Manin-Mumford and a weak version of Zilber-Pink (both are theorems). I am going to start with very basic things, give quite a few examples and define/explain all concepts that I am going to use, so I hope that most of the talk will be accessible to a wide range of people including those who have not heard about Diophantine geometry before.

Location: MALL
Title: Indestructibility and $C^{(n)}$-supercompact cardinals
In the 70's Laver showed that a supercompact cardinal $\kappa$ may be made indestructible by a suitable class of forcings—namely, after a preparatory forcing, the supercompactness of $\kappa$ will not be destroyed by any further $<\kappa$-directed closed forcing. Many indestructibility results have since been written, as well as those demonstrating the impossibility of indestructibility (or even preservation) of many large cardinals. In this talk we will consider the case of $C^{(n)}$-supercompact cardinals—a stronger and more slippery variant of supercompact cardinals—and how they can be made indestructible for $n\leq 2$.

Location: MALL 1
Title: Purity in chains of modules

The model theory of modules extends naturally to certain functor categories. One such category is that of representations of the biinfinite quiver $A_{\infty}^{\infty}$, where each object can be thought of as a biinfinite chain of $R$-modules. This raises the question of how the objects and morphisms of (model theoretic) interest for this category relate to those of $\operatorname{Mod}\ R$. In the simplest case, we take $R$ to be von Neumann regular.

Location: MALL
Title: How far is almost strong compactness from strong compactness.
Almost strong compactness of $\kappa$ can be characterized as follows: for every $\delta < \kappa < \lambda$, there is an elementary embedding $j_{\delta,\lambda}: V \rightarrow M$ with critical point $\geq \delta$, so that $j_{\delta,\lambda}" \lambda \subseteq D \in M$ and $M \vDash |D|< j_{\delta,\lambda}(\kappa)$. Boney and Brooke-Taylor were wondering whether almost strong compactness is essentially the same as strong compactness. Recently, Goldberg showed that if $\kappa$ is of uncountable cofinality and SCH holds from below then these two closely related concepts are the same. In this joint work with Zhixing You, we show that these two can be different in general cases.