Generalized Baire spaces, structures and combinatorics

In this project the complexity of theories is studied. A theory can be used to answer questions of other theories, this provides a notion of complexity. The more general is a theory, more complex it is. A question in one theory can be translated to a question in another theory and then answered. If all the questions on the theory A can be translated to a question on the theory B, then B is more complex than A. In generalised descriptive set theory this notion of complexity can be studied by the use of functions as the translations. Since the questions in a theory is answered by its models, the functions used in generalised descriptive set theory are the functions that maps models of one theory into models of another theory. Another notion of complexity has been provided by model theory, the more models a theory has, more complex it is. The biggest issue with this notion of complexity comes when two theories with the same amount of models are compared, when this happens, this notion of complexity cannot say much about the complexity of the theories involved. In this project the generalised descriptive set theory notion of complexity is developed in more detail to determine their differences with the model theoretic notion of complexity. It has been conjectured that the generalised descriptive set theory notion of complexity is a refinement of the model theoretic notion of complexity, some results have showed that for many theories the complexity given by the model theory notion coincide with the complexity given by the generalised descriptive set theory notion. In the model theoretic notion of complexity, the theories are classified in different groups, classifiable theories, unstable theories, etc, being classifiable the least complex theories. By showing how to construct functions from the models of a classifiable theory to the models of a not classifiable theory, the conjecture would be proved. Generalised descriptive set theory is a young area. Only until 2014 the area was properly establish, when Friedman, Hyttinen, and Kulikov published the monograph Generalised descriptive set theory and classification theory" in which they solve fundamental problems of the area by providing the right fundamental definitions that allowed the generalization of the classic descriptive set theory. Besides develop the generalised descriptive set theory, this project will show a deep connection between model theory and generalised descriptive set theory, it will provide a new perspective about how to generalize the notions of classifiable theories.

Friedman-Hyttinen-Kulikov's conjecture was proved to be true. During this project the relation between greneralized descriptive set theory and model theory was studied. The complexity of different mathematical theories was studied from the point of view of descriptive set theory (Borel reducibility) and Shelah's classification theory. Two different methods to classify first-order countable complete theories developed in the 1980's, the former one from set theory and the latter one from model theory. It was showed that the Borel reducibility complexity is a refinement of the classification theory complexity. During this project, the theory of generalized descriptive set theory was developed further and new notions and objects were found in both fields (set theory and model theory). In general, we can compare the complexity of any two theories from a set theory point of view and it does not contradicts the complexity from model theory. A particular example is the vector spaces vs the order of the reals (the real line). According to both methods, the real line is more complex than the vector spaces.