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# Theory and application of definable forcing

## Cel

Set theory offers very general and powerful methods to prove mathematical theorems. In some instances, set theoretic methods actually prove that certain mathematical questions are undecidable, i.e. neither provable nor refutable. A famous example is the Continuum Hypothesis. Usually such independence proofs use the method of forcing. Forcing has been developed into a deep and sophisticated theory. Especially important are iterations of forcing constructions. The most common iterations are finite and countable support, and the central concept for countable support iteration is Shelah's notion of proper forcing. For strong results in forcing theory (which are important for applications) often definability of the forcing is required . Non elementary proper (ne p) is an especially promising class of such forcing. The development of forcing theory is important for applications to pure mathematics, especially measure theory, algebra and topology. For example, in 2004 we proved a result in forcing theory that was used to prove the undecidability of von Weizsaecker's problem (For every function f from the reals to the reals there is a set X of positive outer measure such that f restricted to X is continuous).
We plan to work on the following questions in connection with nep forcing:
(1) Find preservation theorems.
(2) Develop a theory of non-wellfounded iterations.
(3) Find alternative wellfounded iterations (free limit, non-c.s.i.).
(4) Develop a theory of non-Cohen oracle ccc forcings.
The project will provide m e with advanced training in forcing theory, descriptive set theory and model theory. The research will be conducted at the Einstein Institute at the Hebrew University of Jerusalem, a leading institution in Mathematics. The scientist in charge of the of the project is Saharon Shelah, a worldwide leading set theorist.

### Temat(-y)

FP6-2004-MOBILITY-5
Zobacz inne projekty w ramach tego zaproszenia

### System finansowania

EIF - Marie Curie actions-Intra-European Fellowships

### Koordynator

THE HEBREW UNIVERSITY OF JERUSALEM