Forcing for set- and model-theory

The research field of the project is mathematics, more specifically: logics, even more specifically: set theory and model theory. In mathematics, one tries (amongst other things) to prove or contradict mathematical statements. Logic investigates (amongst other things): What exactly is a proof? Are there statements that can be neither proved nor refuted (in the standard axiomatization of mathematics, ZFC)? It turns out that there are such statements, a famous example is the continuum hypothesis (CH): Every infinite subset of the reals is as large as the reals or as large as the natural numbers. CH belongs to the field of set theory, as does forcing, which is the method to show that CH is neither provable nor refutable. Model theory investigates the connection between formal / logical properties of a theory and the structure of the models of a theory. In particular the project deals with classification theory, which looks for suitable (in particular, natural and useful) dividing lines between chaotic and manageable theories. We investigate a subfield of classification theory which uses forcing: Let us call a theory universal manageable if it necessarily has universal models of many cardinalities. By necessarily, we mean in all forcing extensions. This restriction is necessary, as a theory could have many universal models incidentally for cardinal arithmetic reasons. For example, if a generalisation of CH holds, then every theory has many universal models, but this is not a property of the theory but an artefact of cardinal arithmetic. We can get rid of the artefact by requiring many universal models not only in the current universe, but in all forcing extensions. We conjecture that universal manageable is a natural and useful dividing line, and the project will investigate this notion for several concrete theories.

The topic of the project was mathematical logic, in particular set theory and model theory. Set theory is the usual way to universally axiomatise mathematics, and provides tools to show that some statements are neither provable nor refutable. The most important is forcing: It allows us to extend the current mathematical universe to a bigger one in which we can arrange certain statements to hold. The most prominent example for this effect is the Continuum Hypothesis. Model theory investigates how properties of a theory T (such as the group axioms or the vector space axioms over a fixed field) correspond to the structure of the models of T. Some model theoretic results have the form of a dichotomy: Either T is "nice" and models can be described by a small number of meaningful invariants (such as dimension in case of vector spacecs); or T is chaotic and has many nonisomorphic models. An example for such a result is Shelah's Main Gap theorem. However, in our setting we cannot expect a chaotic T to have many nonisomorphic models in our current universe; rather we hope to show that it does so in a forcing extension. We worked on the model theoretic questions concerning theories T over a predicate P. The ultimate long term goal is to prove a result analogous to the Main Gap: Either T has a nice structure theory (over P), or (in a forcing extension) T has many nonisomorphis models with the same P part. We succeeded to get several results in this direction: On the one hand, the stability part, we showed that if a model of T is stable over P, then types weakly orthogonal to P are quantifier-free definable and have a good notion of independence; and the class of T models admits stable amalgamation. We introduced the notion of full stability over P, and showed that T has prime models in the category of saturated models, and, if T is countable, it has the Gaifman property. On the other hand we continued the investigation of forcing theory, giving some new constructions and answering some set theoretic questions in the process: For example: Let be inaccessible, and B be the Boolean algebra of the subsets of of size < . Then CH at implies that there is a nowhere trivial automorphism of B; it is consistent with any value of 2^ that there is a nowhere trivial automorphism; but it is also consistent that every automorphism is somewhere trivial.