Mathematical Sciences: Certain Set-Theoretic Principles and Their Applications

9505098科兹米德由保罗·科恩发明的强迫方法将戈德尔的结果带入了主流数学领域：关于Banach空间、勒贝格测度或怀特黑德群的一些问题是不可判定的。由于连续统假设似乎与勒贝格积分的基本性质相去甚远，这些原理似乎远离数学实践。然而，就像在连续统假设的情况下，它们影响了现代数学的标准对象。集合论者正在开发一个无限组合原理网络，为原本无法解决的问题提供答案。其中一些原理涉及两个基数组合对象的存在(例如，Morass，或其存在等价于Morass的集合族)；这些对象可以被视为基数的推广。虽然采用这种结构的结构通常比通常的超限归纳结构更复杂，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推广和发展新的规范方法，将Velleman的简化Morass应用于构建布尔代数、拓扑学和其他领域的结构。第二个目标是开发利用两基数组合原理构造强迫概念的新方法。这些强迫概念直接用于证明不可决定性。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当于高间隙Mores的相对可管理的原理，并发展将它们应用于布尔代数、拓扑学和其他领域的方法。Koszmider从不同领域选择了一些著名的公开问题作为测试用例，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、精确和抽象的数学问题。数学机器为这些问题提供解决方案，并给出蓝图(称为定理)，为给定的假设提供结论。证明定理是数学的全部内容；定理直接或间接地影响我们表述和解决科学问题的方式。证明定理是完美的方法吗？不是的！1930年，戈德尔证明，在任何合理的数学体系中，除非我们在数学基础上增加新的公理，否则将会有永远不会被确定的猜想和永远不会被解决的问题。这些无法解决的问题仅仅是人为的逻辑结构吗？不是的！1964年，保罗·科恩发明了强迫方法，从那时起，这种方法就被用来证明许多主流数学问题的不可解性。我们需要知道哪些问题是不可解的，以及数学基础(新公理)中的哪些额外假设使它们是可解的。根据他在现场的工作和部分研究，科兹米德推测，某些组合原理将提供一种新的方法来证明大量重要问题的不可解性。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不同部分，如无限组合数学、布尔代数和拓扑学。科兹米德计划将这一基本主题正式化，并证明其与经典原则的关系，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. ***