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?