Forcing, creatures, oracles and large continuum

The theme of the project is the following set theoretical question: Find forcing constructions that result in large continuum without adding Cohen reals. In particular, we investigate mixed limit creature forcing and oracle/preparatory forcings, as well as connections to idealized forcing. We plan to apply such constructions to questions in the following fields: cardinal characteristics of the continuum, the (dual) Borel Conjecture and (point set) topology.

The topic of the project is set theory. Similar to Euclids axiomatization of Geometry more than 2000 years ago, set theory provides an axiomatization of all of modern mathematics: Nowadays, a mathematical statement is generally accepted to be proven exactly if it can be formally proven in set-theoretic axiom system ZFC. Certain statements can neither be proven nor disproven in ZFC, they are called undecidable. Famous examples are the consistency of ZFC (according to the incompleteness theorem), and the Continuum Hypothesis (the statement: every infinite set of reals has a 1-1 correspondence to either the natural numbers or the reals). Set theory provides methods to prove that such statements are undecidable. The most important method is forcing. Since its development by Cohen in the 1960s it has been expanded into a rich and deep theory. The project contributes to the development of forcing theory. In particular, we ask: How can one increase the continuum in a forcing iteration without adding Cohen reals? In course of the projects several scientific articles were published, and a PhD thesis was successfully completed.