9505098 Koszmider The method of forcing invented by Paul Cohen brought Godel's result into the realm of mainstream mathematics: some problems on Banach spaces, Lebesgue measure, or Whitehead groups are undecidable. As the continuum hypothesis seems far from the basic properties of Lebesgue integral, these principles seem far away from mathematical practice. However, as in the case of the continuum hypothesis, they affect the standard objects of modern mathematics. Set-theorists are developing a network of infinitary combinatorial principles that provide answers to otherwise unsolvable problems. Some of these principles involve the existence of two-cardinal combinatorial objects (e.g., morasses, or families of sets whose existence is equivalent to the existence of morasses); these objects may be viewed as generalizations of cardinals. Although constructions which employ these structures are usually more complicated than constructions by the usual transfinite induction, they may be necessary in more complicated situations. The first objective of the project is to extend and develop new canonical methods of applying Velleman's simplified morasses for constructing structures in Boolean algebras, topology and other fields. A second objective is to develop new methods of constructing forcing notions using two-cardinal combinatorial principles. These forcing notions are used directly for proving undecidability. If the new combinatorial methods are applied to forcing theory, this may result in a series of new consistency results in pure set theory as well as in its applications. The third objective is to introduce relatively manageable principles equivalent to higher gap morasses and develop methods of applying them to the theory of Boolean algebras, topology and other fields. Koszmider has selected well-known open problems from various fields as test cases, conjecturing that some of these can be solved using the above new methods. Mathematics is the language of science. Scientific problems are translated into formal, precise and abstract mathematical problems. Mathematical machinery provides solutions to these problems and gives blueprints (known as theorems) that provide conclusions to given assumptions. Proving theorems is what mathematics is all about; theorems directly or indirectly affect the way we formulate and solve scientific problems. Is proving theorems a perfect method? No! In 1930, Godel showed that in any reasonable system of mathematics there will be conjectures which will never be decided and probems that will never be solved, unless we add new axioms in the foundations of mathematics. Are these unsolvable problems only artificial logical constructions? No! In 1964, Paul Cohen invented the method of forcing, which since then has been used for demonstating unsolvability of many problems of mainstream mathematics. We need to know what problems are unsolvable and what extra assumptions in the foundations of mathematics (new axioms) make them solvable. Based on his work in the field and on partial research, Koszmider conjectures that certain combinatorial principles will provide a new method for demonstrating unsolvability of a large collection of important problems. These principles could thus serve as extra axioms. These problems arise in various parts of mathematics, such as infinitary combilatorics, Boolean algebra, and topology. Koszmider plans to formalize this underlying theme and to prove its relation to classical principles in order to use them efficiently. ***
