Squares, scales and stationary Reflection
Squares, scales and stationary Reflection
复制标题
正方形、刻度和静止反射
DOI：
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发表时间：
2001
影响因子：
0.9
通讯作者：
M. Magidor
中科院分区：
文献类型：
--
作者：
J. Cummings;M. Foreman;M. Magidor
自从哥德尔和科恩的工作表明希尔伯特的第一个问题（连续统假设）独立于数学的通常假设（由Zermelo-Fraenkel集合理论与选择公理（Axiom of Choice， ZFC）公理化）以来，在数学的许多领域出现了无数的独立结果。这些结果导致了对几个组合原理的系统研究，这些组合原理已被证明在解决许多重要的独立陈述方面是有效的。其中最突出的是詹森发现的菱形（招收）和方形（招收）原理。同时，人们也尝试通过大基数公理或反射公理来寻找合适的ZFC自然增强。这两个方向之间存在紧张关系，因为詹森原理倾向于暗示一个相当严格的数学宇宙，与反射特性不一致。第三个发展是希拉发现的“PCF理论”，这是一种基数算术的概括，主要是在ZFC内部确定的。在本文中，我们考虑了这三种理论在奇异基数的背景下的相互作用，重点关注了正方形和尺度之间的各种含义（PCF理论中的一个基本概念），以及相对强形式的正方形和平稳集反射之间的一致性结果。
Since the work of Godel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales (a fundamental notion in PCF theory), and on consistency results between relatively strong forms of square and stationary set reflection.