Mathematics

We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP₁-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for $\omega$-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are $\omega$-categorical.

An important dividing line in the class of unstable theories is being NSOP₁, which is more general than being simple. In NSOP₁ 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 NSOP₁ 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 NSOP₁ and various other properties. We prove that, in a thick positive NSOP₁ theory, Kim-independence over existentially closed models has all the nice properties that it is known to have in a first-order NSOP₁ theory. We also provide a Kim-Pillay style theorem, characterising which thick positive theories are NSOP₁ 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 finite height.

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.

Neostability focuses on developing the powerful tools and ideas for stable theories in broader, less well-behaved, classes of theories. This has been hugely successful in simple theories and more recently in NSOP₁ theories. A central notion in this context is that of an independence relation, which in stable and simple theories comes from forking and in NSOP₁ theories it comes from Kim-forking. There are many mathematically interesting structures that cannot be studied in the classical framework of first-order logic, but where ideas from neostability, such as independence, apply. For this reason, neostability has been developed in more general logical frameworks, such as continuous logic, positive logic and a very general category-theoretic approach that subsumes all these frameworks. In this talk I will focus on the categorical approach to independence relations, and how even in this general setting we get canonicity theorems that are surprisingly close to those we know from first-order logic. I will finish with recent examples of where the categorical framework can be applied.

Independence relations play a central role in model theory. We all already know examples of independence relations, for example linear independence in vector spaces. Shelah generalised the idea of independence, through the notion of forking, to a very broad class of mathematical structures: the so-called stable structures. In vector spaces (which are stable structures) this forking independence then coincides with linear independence. This is no coincidence, because it turns out that independence is canonical. That is, in any structure there is at most one nice enough independence relation, and if it exists it coincides with forking independence. Later, Kim and Pillay generalised work on forking independence to the even broader class of simple structures. This does come at the cost of a certain property that forking has in stable structures, but we keep enough properties for a canonicity theorem. Going one step further there is the class of NSOP₁ structures, which is more general than the class of simple structures. Kaplan and Ramsey were able to generalise the work on independence to this NSOP₁ class. This comes at the cost of another property, but we still keep canonicity (technical note: we now have to replace forking by Kim-forking).

All of the above takes place in the framework of first-order logic. Many interesting mathematical structures do not fit well in this framework. In this talk we will have a look at results that generalise these theorems on canonicity of independence to a very general category-theoretic framework, called Abstract Elementary Categories (AECats). That is, after translating the properties that we know independence has in stable / simple / NSOP₁ structures to our category-theoretic framework, we get a definition of a stable / simple / NSOP₁-like independence relation. The main results then state that there can be at most one stable / simple / NSOP₁-like independence relation on an AECat and that it has to be given by an appropriate generalisation of forking (or Kim-forking for NSOP₁-like independence).

In Shelah's stability hierarchy we classify theories using combinatorial properties. Some important classes are: stable, simple and NSOP₁, 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 NSOP₁-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.

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 NSOP₁ 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 NSOP₁-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.

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.

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.

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example: linear independence in vector spaces yields such a relation.

Some important classes in the stability hierarchy are stable, simple and NSOP₁, each being contained in the next. For each of these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example: simplicity is equivalent to admitting a certain independence relation, which must then be unique.

All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic and even a very general category-theoretic framework. In this thesis we continue this work.

We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple or NSOP₁-like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.

Switching frameworks, we generalise Kim-dividing in NSOP₁ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP₁ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.

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).