Papers and preprints.

There is nothing yet of this type...

Talks and the corresponding material.

**A category-theoretic version of the Kim-Pillay theorem**Slides used for my talk at the BPGMTC20 and the British Logic Colloquium 2019.Slides (PDF) • Expand abstractCollapse abstractIn 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.

1;2;Miscellaneous notes and pieces of mathematics.

There is nothing yet of this type...

My theses.

**Classifying Topoi and Model Theory**Master's thesis about classifying topoi and connection to model theory. Supervised by Jaap van Oosten, at Utrecht University.PDF • Expand abstractCollapse abstractEvery 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.

1;2;3;**The Logic of Unprovability**Bachelor's thesis about GĂ¶del's incompleteness theorems. Supervised by Jaap van Oosten, at Utrecht University.PDF • Expand abstractCollapse abstractThe 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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