## Final Activity Report Summary - DEFINABLE FORCING (Theory and application of definable forcing)

Background: Set theory is a mathematical discipline that investigates infinite sets in a rather general way. A typical set theoretical question is CH, Hilbert's first problem: Does every infinite set of reals either have the size of the reals or of the integers? Mathematical logic is the part of mathematics that as its object investigates mathematical reasoning itself, so in a way it is meta-mathematics. Concepts such as proof or algorithm are investigated, and it turns out that there are very satisfying mathematical definitions for these a priori rather intuitive concepts. This allows formulating questions such as: Is a certain theorem decidable? Is a function computable? It turns out that set theory is very close to logic (and in fact it is usually considered to be one of the main disciplines of mathematical logic), for several reasons, among them:

(1) Several natural questions in set theory turn out to be undecidable, for example CH.

(2) Set theory actually provides the methods to prove that a wide array of questions from many mathematical fields are undecidable.

(3) Set theory is one (and currently the most common) universal foundation for mathematics: A theorem is generally considered to be proven if it is (or at least could theoretically be) proven in a specific axiomatisation of set theory, ZFC.

Forcing is a central method of set theory: Starting with a mathematical universe (i.e. a structure containing the natural numbers, the reals, to each set its powerset etc, a bit more formally a wellfounded ZFC model) we can add a new "generic" object for a particula partial order to get a new mathematical universe, and we force this new universe to satisfy certain properties (e.g. the negation of CH). This way we can prove that the negation of CH is possible in a mathematical universe, i.e. that CH is not provable.

The Marie Curie project contributed to the development of the theory of forcing, in particular of definable proper forcing. We proved various preservation theorems for countable support iterations (nep forcings that do not make old positive Borel sets null do not make any old positive set null, which is iterable), we showed examples for limitations of such theorems (a Sacks real can appear at stage omega in a proper countable support iteration that does not add any reals at finite stages), we developed new constructions for creature forcings to get large continuum, and applied them to show that you can simultaneously distinguish several well known cardinal characteristics (and also get a perfect set of different simple characteristics defined with a real parameter). We worked on the theory of non-elementary proper forcing; and also investigated the pressing down game (related to precipitous ideals and large cardinals).

(1) Several natural questions in set theory turn out to be undecidable, for example CH.

(2) Set theory actually provides the methods to prove that a wide array of questions from many mathematical fields are undecidable.

(3) Set theory is one (and currently the most common) universal foundation for mathematics: A theorem is generally considered to be proven if it is (or at least could theoretically be) proven in a specific axiomatisation of set theory, ZFC.

Forcing is a central method of set theory: Starting with a mathematical universe (i.e. a structure containing the natural numbers, the reals, to each set its powerset etc, a bit more formally a wellfounded ZFC model) we can add a new "generic" object for a particula partial order to get a new mathematical universe, and we force this new universe to satisfy certain properties (e.g. the negation of CH). This way we can prove that the negation of CH is possible in a mathematical universe, i.e. that CH is not provable.

The Marie Curie project contributed to the development of the theory of forcing, in particular of definable proper forcing. We proved various preservation theorems for countable support iterations (nep forcings that do not make old positive Borel sets null do not make any old positive set null, which is iterable), we showed examples for limitations of such theorems (a Sacks real can appear at stage omega in a proper countable support iteration that does not add any reals at finite stages), we developed new constructions for creature forcings to get large continuum, and applied them to show that you can simultaneously distinguish several well known cardinal characteristics (and also get a perfect set of different simple characteristics defined with a real parameter). We worked on the theory of non-elementary proper forcing; and also investigated the pressing down game (related to precipitous ideals and large cardinals).