Connections between model theory and set theory

We propose a project in the field of mathematical logic which aims at the investigation of set theoretic aspects of model theoretic notions. The project will be based on previous research of ours and will be composed of two parts on which we plan to work simultaneously. The first part, continuing the topic of my PhD thesis, concerns the relationship between two different notions of complexity for classes of countable structures, namely depth of an -stable classifiable theory (according to Shelah), and the descriptive set theoretic notion of Borel-reducibility. In previous work, we have established a correspondence between depth 1 theories and "smooth" classes, we have found "cofinal" sequences of theories whose complexity simultaneously increases with respect to both notions, and we have found a surprising example of depth 2 which has a non-Borel isomorphism relation. Continuing in this line, we plan to investigate several questions that remain open, the most outstanding of which are the precise Borel reducibility-degree of our non- Borel example (in particular the question whether that degree is maximal), the establishment of a close correspondence between the two notions of complexity under extra-assumptions which exclude the complexity phenomenon of our non-Borel example and the question whether non-trivial essentially countable degrees exist in our context. The second topic concerns the question of how absolute (relative to ZFC) basic notions in the model theory of infinitary logic, specifically L1 ,, are. We focus on the notions of model existence (in a given cardinality) and categoricity. While we have shown that model existence is absolute in Aleph(1) and non-absolute in higher cardinalities (below Beth( 1 )), we plan on finding complete sentences with the same properties and enlarge the range our knowledge of set-theoretic properties that may have an influence on model existence. Also the case of Aleph() is still open, even for incomplete sentences. Concerning categoricity, we plan on working in the direction of showing absoluteness of Aleph(1)-categoricity. The main ingredient is to show that failure of amalgamation for countable models absolutely implies the existence of many models in Aleph(1). We will build on some recent technical advancements of ours in that matter, as well as on recent work of Shelah`s that shows absoluteness of Aleph(1)-categoricity under extra-assumptions. While we can show (using our non-absoluteness results of model-existence) that categoricity is non-absolute in cardinalities between Aleph(2) and Beth(1 ), we plan on working on our conjecture that categoricity should be absolute in cardinalities Beth(1 ) and above.

Mathematical logic is concerned with the foundation of mathematics. Central questions are: how can mathematics be based on a formal system which allows the deduction of all mathematical truths, and how can the consistency of such a system be ensured? During the 20th century, significant progress has been made. At the same time, many sub-areas of mathematical logic became more and more independent and developed their own dynamics. Our main interest is to bring those diverging disciplines closer together to each other again. We concentrate on the classical areas of set theory (which serves as a foundational system of mathematics: all mathematical objects can be constructed as sets) and model theory (which deals with the abstract notions of mathematical objects and of theories). We investigate in particular, in what way model theoretic properties can depend on the surrounding set theoretic universe (the class of all sets as foundation of mathematics). Concretely, we considered the notions of the existence of models (of a given theory) of various sizes (cardinalities) as well as the number of models of a given size. We worked on the long standing open problem of Vaught`s Conjecture (VC) which makes a statement about the number of countable models of a theory. Naturally this classical question remains open, but we significantly cleared up the theoretical base of a possible strategy to solve it (the result: if there is a counter- example to VC, there is also one with no models beyond the first uncountable cardinality). We also provided a new, elegant proof of Harrington`s Theorem (which concerns the complexity of models of a counter-example to VC). In another line, we worked out an abstract method to construct theories which only have models up to a certain size. An additional benefit of this work has been to provide examples of theories with new, until now unknown amalgamation spectra (which concerns the possibility to fuse models of certain cardinalities). Finally we have also been investigating a new phenomenon concerning maximal (non- extendible) models of theories. We showed that it is possible to have maximal models of a theory and yet at the same time other models of even larger cardinality. Such theories have been unknown until now.