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Themes for research and theses

An important Utrecht specialism is Categorical Logic.
Category theory was invented by Eilenberg and MacLane around 1945. It provides an organization of mathematical structures in which, rather than sets with elements, the notion of function (also called map, or morphism) is central.
A category consists of a collection of objects and a collection of morphisms; each morphism "goes" from one object to another. Morphisms can be composed, and this composition is associative. Think of: sets and functions, groups and homomorphisms, topological spaces and continuous maps.
It turns out that many set-theoretic properties can be expressed using only the language of maps and composition. To give a very simple example: a function f:X->Y is injective precisely if for every two maps g,h:Z->X it holds that whenever the compositions fg and fh:Z->Y are equal, then g=h.
Categorical logic pursues this point of view. It constructs definitions of set-theoretic concepts (like the set of natural numbers, or the power set of a set) in the language of objects and maps, and gives a very general notion of a model of a theory in a category. For example, a model of the theory of groups in the category of topological spaces is a topological group (a topological space with a group structure such that the group operations are continuous).
An important type of categories are topoi. A topos is a category in which most constructions one can perform with sets (like, form the power set), are possible ; it is a "nonstandard universe of sets". However, the logic which governs these universes is intuitionistic logic, where one is not allowed to use the principle of excluded third in arguments. An example of a topos is the category of sheaves on a topological space. Another kind of topoi have been defined using notions from Recursion Theory.
Topos Theory is also studied in theoretical Computer Science. Recently, there has even been interest in topoi within theoretical physics.
A recent master thesis was about connections between topos theory and classical model theory.

Another recent master thesis is about nonstandard models of arithmetic. Peano Arithmetic is an axiom system which aims to characterize basic algebraic properties of the natural numbers. However, there are also nonstandard models of this theory, models in which there exist nonstandard, "infinitely big" natural numbers. Now suppose in such a nonstandard model M we consider a nonstandard prime number p. Since M satisfies the axioms of elementary number theory, in M we can calculate "modulo p", and we get a field. This field has characteristic 0. A question is then: which fields arise in this way? There are nice results which relate this to the question which induction axioms are true in M.
Another pertinent question in this area is: how many induction axioms are necessary to prove that there exist infinitely many primes?


 
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