Papers that are (intended to be) submitted to a journal.

With Jan Dobrowolski, preprint, 17 May 2021.arXiv • Expand abstractCollapse abstract

An important dividing line in the class of unstable theories is being NSOP_{1}, which is more general than being simple. In NSOP_{1} theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced by another independence notion, called Kim-independence. We generalise Kim-independence over models in NSOP_{1} theories to positive logic—a proper generalisation of first-order logic where negation is not built in, but can be added as desired. For example, an important application is that we can add hyperimaginary sorts to a positive theory to get another positive theory, preserving NSOP_{1} and various other properties. We prove that, in a thick positive NSOP_{1} theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in a first-order NSOP_{1} theory. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP_{1} by the existence of a certain independence relation. Furthermore, this independence relation must then be the same as Kim-independence. Thickness is the mild assumption that being an indiscernible sequence is type-definable.

In first-order logic Kim-independence is defined in terms of Morley sequences in global invariant types. These may not exist in thick positive theories. We solve this by working with Morley sequences in global Lascar-invariant types, which do exist in thick positive theories. We also simplify certain tree constructions that were used in the study of Kim-independence in first-order theories. In particular, we only work with trees of

Preprint, 8 May 2020.arXiv • Expand abstractCollapse abstract

We construct a 2-equivalence $\mathfrak{CohTheory}^\text{op} \simeq \mathfrak{TypeSpaceFunc}$. Here $\mathfrak{CohTheory}$ is the 2-category of positive theories and $\mathfrak{TypeSpaceFunc}$ is the 2-category of type space functors. We give a precise definition of interpretations for positive logic, which will be the 1-cells in $\mathfrak{CohTheory}$. The 2-cells are definable homomorphisms. The 2-equivalence restricts to a duality of categories, making precise the philosophy that a theory is 'the same' as the collection of its type spaces (i.e. its type space functor).

In characterising those functors that arise as type space functors, we find that they are specific instances of (coherent) hyperdoctrines. This connects two different schools of thought on the logical structure of a theory.

The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

Papers that have been published or are accepted.

We introduce the framework of AECats (abstract elementary categories), generalising both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory ("cat", as introduced by Ben-Yaacov) forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of (subsets of) models of a positive or continuous theory is an AECat.

The Kim-Pillay theorem for first-order logic characterises simple theories by the properties dividing independence has. We prove a version of the Kim-Pillay theorem for AECats with the amalgamation property, generalising the first-order version and existing versions for positive logic.

Talks and the corresponding material.

Talk at the Helsinki Logic Seminar.Slides (PDF) • Expand abstractCollapse abstract

In Shelah's stability hierarchy we classify theories using combinatorial properties. Some important classes are: stable, simple and NSOP_{1}, each being contained in the next. We can characterise these classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Accessible categories are a very general framework. The category of models of some theory is an accessible category, every AEC (abstract elementary class) is an accessible category, but even then accessible categories are more general. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP_{1}-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Slides for a talk at SMTH2.Slides (PDF) • Expand abstractCollapse abstract

This is joint work with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where negation is not built in, but can be added as desired. In this talk I will give a brief introduction to positive logic. We will have a look at the challenges that positive logic presents us and how they can be solved. I will also explain why positive logic is useful and how we can use it to study structures that do not fit the usual framework of first-order logic.

An important dividing line in the class of unstable theories is being NSOP1, which is more general than being simple. There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence, generalising forking independence from stable and simple theories. We have generalised work on Kim-independence to the setting of positive logic. I will assume no prior knowledge of this topic and will introduce the definitions of NSOP1 and Kim-independence, and how we can make sense of this in positive logic. Our results can be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-independence.

Talk at the Masaryk University Algebra Seminar.Slides (PDF) • Youtube • Expand abstractCollapse abstract

In Shelah's classification of first-order theories we classify theories using combinatorial properties. The most well-known class is that of stable theories, which are very well-behaved. Simple theories are more general, and then even more general are the NSOP_{1} theories. We can characterise those classes by the existence of a certain independence relation. For example, in vector spaces such an independence relation comes from linear independence. Part of this characterisation is canonicity of the independence relation: there can be at most one nice enough independence relation in a theory.

Lieberman, Rosický and Vasey proved canonicity of stable-like independence relations in accessible categories. Inspired by this we introduce the framework of AECats (abstract elementary categories) and prove canonicity for simple-like and NSOP_{1}-like independence relations. This way we reconstruct part of the hierarchy that we have for first-order theories, but now in the very general category-theoretic setting.

Slides for a talk at GSCL XXII.Slides (PDF) • Expand abstractCollapse abstract

This is joint work (in progress) with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where we only allow the logical connectives for conjunction, disjunction, falsehood, truth and the existential quantifier. For example, if we add hyperimaginaries as parameters to our monster model then we leave the framework of first-order logic, but we remain in the framework of positive logic. Another interesting example is the positive theory of existentially closed exponential fields (ECEF), introduced by Haykazyan and Kirby.

There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence. We have generalised the work on Kim-independence to the setting of positive logic. In this talk I will explain the challenges that positive logic presents us and how we solve them. Our result can then be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-dividing.

Haykazyan and Kirby proved that ECEF has a nice enough independence relation and showed that it is thus NSOP1. So our theorem applies and confirms that this independence relation is given by Kim-dividing. The operation of adding hyperimaginaries as parameters preserves the NSOP1 property. So in particular, if we start with an NSOP1 first-order theory and we add hyperimaginaries then our theorem still applies.

Slides for a talk at the logic seminar at the Hebrew University of Jerusalem.Slides (PDF) • Expand abstractCollapse abstract

This is joint work (in progress) with Jan Dobrowolski. Positive logic is a proper generalisation of first-order logic where we only allow the logical connectives for conjunction, disjunction, falsehood, truth and the existential quantifier. For example, if we add hyperimaginaries as parameters to our monster model then we leave the framework of first-order logic, but we remain in the framework of positive logic. Another interesting example is the positive theory of existentially closed exponential fields (ECEF), introduced by Haykazyan and Kirby.

There has been a lot of recent work on the class of NSOP1 theories for first-order logic. The natural independence relation in this class is given by Kim-independence. We have generalised the work on Kim-independence to the setting of positive logic. In this talk I will explain the challenges that positive logic presents us and how we solve them. Our result can then be summarised as a Kim-Pillay style theorem: a thick positive theory is NSOP1 if and only if there is a nice enough independence relation, and in this case the independence relation is given by Kim-dividing.

Haykazyan and Kirby proved that ECEF has a nice enough independence relation and showed that it is thus NSOP1. So our theorem applies and confirms that this independence relation is given by Kim-dividing. The operation of adding hyperimaginaries as parameters preserves the NSOP1 property. So in particular, if we start with an NSOP1 first-order theory and we add hyperimaginaries then our theorem still applies.

Poster for the Topics in Category Theory 2020 spring school.Poster (PDF) • Expand abstractCollapse abstract

For a first-order theory $T$ we can collect the type spaces $\operatorname{S}_n(T)$ in a contravariant functor between the category of finite sets and the category of Stone spaces. Haykazyan generalised this idea to positive theories, replacing Stone spaces by spectral spaces. We characterise those functors that arise as a positive type space functor. This results in a duality between the category of positive theories with strong interpretations and the category of type space functors. Using the Stone duality between spectral spaces and distributive lattices, we can alternatively view type space functors as functors into the category of distributive lattices. This gives another characterisation of the same functors, namely as specific instances of coherent hyperdoctrines. The key ingredient, the Deligne completeness theorem, arises from topos theory, where positive theories have been studied under the name of coherent theories.

Slides used for my talk at the BPGMTC20 and the British Logic Colloquium 2019. These slides are about an older version of the paper

In Shelah's classification of first-order theories using combinatorial properties, the notion of a stable theory is the most well-known. Stable theories are very well-behaved. The notion of a simple theory is a generalisation of this. The theory of the random graph is the prototypical example of a simple theory. In particular, one can develop the concept of forking independence in these theories. This is a generalisation of linear independence in vector spaces, for example. The Kim-Pillay theorem gives us a way to characterise simple theories based on the existence of a certain independence relation. It states that if an independence relation of a certain form exists, the theory is simple and that this independence relation coincides with forking independence. For example, for the random graph this can be applied to the independence relation that says that sets $A$ and $B$ are independent over $C$ if $A \cap B \subseteq C$. We will recall all this in more detail at the start of the talk. After that, we will set up a category-theoretic framework where we can make sense of certain model-theoretic tools and definitions. Taking inspiration from work of Lieberman, Vasey and Rosický, we can define what an independence relation is in this framework. This then allows us to formulate a category-theoretic version of the Kim-Pillay theorem. We will finish by looking at a few examples of where this framework applies, including positive logic and continuous logic. In particular, we will recover the original Kim-Pillay theorem as one of these applications.

Miscellaneous notes and pieces of mathematics.

Short note translating some standard tools in model theory to AECats. PDF • Expand abstractCollapse abstract

In this note we translate standard tools and arguments in model theory to the framework of finitely short AECats. In particular we prove that a form of compactness holds. From this we then get some standard tools for manipulating and building indiscernible sequences.

My theses.

Master's thesis about classifying topoi and connection to model theory. Supervised by Jaap van Oosten, at Utrecht University.PDF • Expand abstractCollapse abstract

Every geometric theory has a classifying topos, but when trying to extend this to full first-order theories one may run into trouble. Such a first-order classifying topos for a first-order theory $T$, is a topos $\mathcal{F}$ such that for every topos $\mathcal{E}$, the models of $T$ in $\mathcal{E}$ correspond to open geometric morphisms $\mathcal{E} \to \mathcal{F}$. The trouble is that not every first-order theory may have such a first-order classifying topos, as was pointed out by Carsten Butz and Peter Johnstone. They characterized which theories do admit such a first-order classifying topos, and show how to construct such a first-order classifying topos.

The work of Butz and Johnstone is the main subject of this thesis. The construction of a classifying topos for both geometric theories and first-order theories is worked out in detail. We will also study the characterization of which theories admit a first-order classifying topos. In doing so, we obtain certain completeness results that are interesting in their own right. These are completeness results for deduction-systems for various kinds of infinitary logic, with respect to models in topoi. Building on top of those results, we also obtain a completeness result for classical infinitary logic, with respect to Boolean topoi.

One of the first goals of this thesis was to form a link between Topos Theory and Model Theory, via the first-order classifying topos. In order to bridge the gap between the intuitionistic logic of topoi and the classical logic of Model Theory, we introduce the concept of a Boolean classifying topos. We provide a characterization of which first-order theories admit such a Boolean classifying topos, much like the one for first-order classifying topoi. Then we give a simple example of how to link Boolean classifying topoi to Model Theory, by characterizing complete theories in terms of their Boolean classifying topos.

Bachelor's thesis about GĂ¶del's incompleteness theorems. Supervised by Jaap van Oosten, at Utrecht University.PDF • Expand abstractCollapse abstract

The first incompleteness theorem of Kurt Gödel states that a theory in which we can develop most of modern arithmetic is incomplete. We will take a look at such a theory: Peano Arithmetic (PA). We will develop the tools to formulate a sentence that essentially asserts in PA that it is not provable. Then we use this sentence to prove both Gödel's first and second incompleteness theorems, where the second states that PA cannot prove its own consistency. To prove Gödel's first incompleteness theorem also for consistent extensions of PA we will use Rosser sentences. These are sentences that are equivalent in PA to the assertion that a disproof of them occurs before any proof of them. After developing the necessary technical tools, we prove that Rosser sentences need not to be all provably equivalent but there are constructions where they are *(after a paper by Guaspari and Solovay)*.

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