New developments regarding forcing in set theory

We investigate forcing constructions to get large continuum for "set theory of the reals" applications, as well as forcing and large cardinals: tree properties, ideals and reflection.

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 deals in particular with cardinal characteristics. A typical example: The union of countably many null sets is null. Of course there are continuum many null sets (e.g., all singletons) whose union is positive. So how many nullsets are required to get a non-null set? The answer (a cardinal characteristic) is called the additivity of null, add(null). So 0 < add(null) = 20 , and under CH add(null) = 20 = 1 . Using the ideals of Lebesgue-null and meager, one can define several other cardinal characteristics, which are summarized in Cichons diagram. One result of the project was that many of these characteristics can be simultaneously different.