Mathematical Sciences: Certain Set-Theoretic Principles and Their Applications

kozmider Paul Cohen发明的强迫方法将哥德尔的结果带入了主流数学的领域：Banach空间、Lebesgue测度或Whitehead群上的一些问题是不可确定的。由于连续统假设似乎与勒贝格积分的基本性质相距甚远，这些原理似乎与数学实践相距甚远。然而，就像连续统假设的情况一样，它们影响了现代数学的标准对象。集合论者正在发展一个由无限组合原理组成的网络，为其他无法解决的问题提供答案。其中一些原理涉及到双基数组合对象的存在性（例如，泥沼，或其存在性等同于泥沼的存在性的集合族）；这些对象可以看作是基数的概括。虽然使用这些结构的结构通常比通常的超限归纳法的结构更复杂，但在更复杂的情况下它们可能是必要的。该项目的第一个目标是扩展和发展新的规范方法，应用Velleman的简化泥沼来构造布尔代数、拓扑和其他领域的结构。第二个目标是发展新的方法构造强迫概念使用二基数组合原则。这些强迫概念直接用于证明不可判定性。如果将新的组合方法应用于强迫理论，可能会在纯集合论及其应用中产生一系列新的一致性结果。第三个目标是引入相对易于管理的原理，相当于高间隙沼泽，并开发将其应用于布尔代数，拓扑和其他领域的理论的方法。Koszmider从各个领域中选择了一些著名的开放问题作为测试用例，并推测其中一些问题可以使用上述新方法来解决。数学是科学的语言。科学问题被转化为形式的、精确的和抽象的数学问题。数学机制为这些问题提供了解决方案，并给出了为给定假设提供结论的蓝图（称为定理）。证明定理是数学的全部；定理直接或间接地影响着我们表述和解决科学问题的方式。证明定理是一个完美的方法吗？不!1930年，哥德尔指出，在任何合理的数学体系中，都存在永远无法确定的猜想和永远无法解决的问题，除非我们在数学基础中添加新的公理。这些无法解决的问题仅仅是人为的逻辑建构吗？不!1964年，保罗·科恩（Paul Cohen）发明了强迫法，从那时起，这种方法就被用来证明主流数学中许多问题的不可解性。我们需要知道哪些问题是无法解决的，以及数学基础中的哪些额外假设（新公理）使它们可以解决。根据他在该领域的工作和部分研究，科兹米德推测，某些组合原理将为证明大量重要问题的不可解性提供一种新的方法。因此，这些原则可以作为额外的公理。这些问题出现在数学的各个部分，如无穷组合学、布尔代数和拓扑学。科兹米德计划将这个潜在的主题形式化，并证明它与经典原理的关系，以便有效地使用它们。＊＊＊

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. ***