Mathematical Sciences: Certain Set-Theoretic Principles and Their Applications

小行星9505098 Paul Cohen发明的强迫方法将哥德尔的结果带入了主流数学领域：Banach空间上的一些问题， 勒贝格测度或怀特黑德群是不可判定的。 作为连续体 假设似乎远离勒贝格积分的基本性质，这些 数学原理似乎离数学实践很远。 然而，正如在 在连续统假设的情况下，它们影响了现代的标准对象， 数学 集合理论家正在开发一个无限网络 组合原理，提供了答案，否则无法解决 问题 其中一些原则涉及到双基数的存在 组合对象（例如，morasses，或家庭的集合，其存在 相当于存在的沼泽）;这些对象可以被视为 Cardinals的分类 尽管采用这些的结构 结构通常比通常的结构更复杂， 超限归纳法，它们可能是必要的，在更复杂的 situations. 该项目的第一个目标是扩大和发展 应用Velleman的简化Morasses的新规范方法 在布尔代数、拓扑学和其他领域构造结构。 一 第二个目标是开发使用两个基数组合原理构造强制概念的新方法。 这些强制性的观念 直接用于证明不可判定性。 如果新的组合 方法应用于强迫理论，这可能会导致一系列新的 一致性的结果在纯集合论以及它的应用。 的 第三个目标是引入相对可管理的原则， 更高的差距沼泽和发展的方法，将其应用于理论 布尔代数、拓扑学和其他领域。 Koszmider选择了 来自各个领域的著名开放问题作为测试用例， 其中一些问题可以通过上述新方法解决。 数学是科学的语言。 科学问题被转化为形式的、精确的、抽象的数学问题。 数学机器为这些问题提供了解决方案，并给出了 蓝图（称为定理），提供结论，以给定 假设。 证明定理是数学的全部;定理 直接或间接地影响我们制定和解决科学问题的方式， 问题 证明定理是一种完美的方法吗？ 不！不！不！ 1930年，哥德尔证明 在任何合理的数学体系中， 永远不会被决定，永远不会被解决的问题，除非我们加上 数学基础中的新公理 这些问题是无法解决的吗 只是人为的逻辑结构？ 不！不！不！ 1964年，保罗·科恩发明了强迫的方法，从那时起， 许多主流数学问题的不可解性。 我们需要知道 哪些问题是无法解决的，基础中有哪些额外的假设， 新的数学公理使它们可解。 基于他在 科兹米德在实地和部分研究中证实， 组合原理将提供一种新的方法， 大量重要问题的不可解决性。 这些原则 因此可以作为额外的公理。 这些问题出现在不同的地方， 数学，如无限combilatorics，布尔代数，和 topology. Koszmider计划将这一基本主题正式化，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. ***