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.