Norms on possibilities I: Forcing with trees and creatures

In this Memoir ¹ we present a systematic study of the method of norms on possibilities of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the solutions we develop various ideas interesting per se. These include a new iterable condition for "not adding Cohen reals" (which has a flavour of preserving special properties of p-points), new intriguing properties of ultrafilters (weaker than being Ramsey but stronger than p-point) and some new applications of variants of the PP-property. The first author thanks the Hebrew University of Jerusalem and the Lady Davis Foundation for the Golda Meir Postdoctoral Fellowship, and KBN (State Committee for Scientific Research, Poland) for partial support through grant 2P03A01109. The research of the second author was partially supported by "Basic Research Foundation" founded by the Israel Academy of Sciences and Humanities. Publication 470.