Mathematical Sciences: Topics in Logic and Category Theory

布拉斯的研究主要集中在三个方面：（1） 组合集合论，特别是基数的研究 连续统的特征;（2）拓扑的逻辑，特别是 拓扑的几何态射之间的连接，内部的 高阶逻辑和指称语义的域;以及 (3)线性逻辑和相关系统的博弈语义学。 集合论提供了一种流行的方法， 所有的数学，最著名的系统化尝试， 罗素和怀特海的《数学原理》 20世纪早期。 重要的基础性 问题涉及的独立性和一致性， 用于建立集合论的公理。 策梅洛和弗伦克尔公理 (ZF是最方便的基本公理集之一。 然而，自从Kurt Goedel在1989年的工作以来， 20世纪30年代，没有公理集理论可以完成，以及 一致 这意味着没有一套公理可以是强大的 足以证明每一个可能的命题或其否定， 但不是两者都有。 在这个惊人的定理上， 理论，处理可能的命题P，使得P或Not P可以加到ZF上而不会导致矛盾。 任何 这样的命题P被认为是与ZF无关的，并且可以采用 作为集合论的附加公理。 主要技术为 找到这样的独立命题是保罗·科恩所谓的 强迫的方法及其后代。 这就是思想的循环 参与并激励大部分调查人员 research.

The research of Andreas Blass is in three areas: (1) combinatorial set theory, particularly the study of cardinal characteristics of the continuum; (2) logic of topoi, particularly connections between geometric morphisms of topoi, the internal higher-order logic, and domains for denotational semantics; and (3) game semantics for linear logic and related systems. Set theory provides the popular way to lay the foundations for all of mathematics, the best known systematic attempt to do this being Russell and Whitehead's Principia Mathematica, dating from the early days of the 20th Century. Important foundational questions concern the independence and the consistency of the axioms used to establish set theory. Zermelo and Fraenkel's axioms (ZF for short) are one of the most convenient sets of basic axioms. However, it has been known since the work of Kurt Goedel in the 1930's that no axioms for set theory can be complete as well as consistent. This means that no set of axioms can be powerful enough to prove every possible proposition or else its negation, but not both. Upon this startling theorem has been erected a rich theory, treating possible propositions P such that either P or Not P can be added to ZF without resulting in a contradiction. Any such proposition P is said to be independent of ZF and can be taken as an additional axiom of set theory. The principal technique for finding such independent propositions is Paul J. Cohen's so-called method of forcing and its offspring. This is the circle of ideas involved in and motivating a large part of the investigator's research.