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INNER MODEL THEORY AND LARGE CARDINALS

内模型理论和大基数

基本信息

批准号：
10640118
负责人：
BRENDLE Jorg
金额：
$ 2.05万
依托单位：
Kobe University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

Research in this project was devoted to inner model theory, large cardinals, forcing theory, cardinal invariants of the continuum and other subfields of set theory, as well as to the interplay between them and to applications to other areas of pure mathematics. We briefly sketch the main topics and results.1.Maximality properties of inner models and elementary embeddings. For example, we investigated under which circumstances the Jonsson property holds for a given cardinal in the core model K if it holds in the universe V.2.Research on infinite time Turning machines. We showed in particular that, if λ is the supremum of the writable ordinals, i.e. ordinals which arise at outputs of computations on input 0, and γ is the supremum of clockable ordinals, i.e. ordinals which are lengths of halting computations on input 0, then λ=γ.3.We proved the set of reals Cohen over a model of ZFC must either be empty or non-meager.4.The effect of cardinal invariants of the continuum on combinatorial properties of uncountable cardinals. For example, we proved that additivity of the null ideal implies that Martin's axiom MA holds for any Cohen algebra. On the other hand, we showed it is consistent that the continuum c is large and covering of the null ideal as well as the combinatorial principle * hold.5.Say that PSP(κ, Γ) holds if every set in the pointclass Γ of size at least κ has a perfect subset. We showed that PSP(ΝィイD21ィエD2, GィイD2ΝィエD2ィイD21ィエD2) holds if and only if σ 【greater than or equal】 ΝィイD22ィエD2 where GィイD2ΝィエD2ィイD21ィエD2 is the class of sets which are intersections of (at most) ΝィイD21ィエD2 many open sets and σ is the dominating number.6.We proved that every maximal confinitary subgroup of Sym(ω), the permutation group of the natural numbers, has size at least the cardinality of the smallest non-meager set reals.7.Completing a cycle of results initiated by Shelah and Spinas, we obtained that if c=ΝィイD22ィエD2 then there is a Gross space over every countably infinite field.

研究在这个项目是专门的内部模型理论，大基数，迫使理论，基数不变的连续和其他子领域的集合论，以及它们之间的相互作用和应用到其他领域的纯数学。我们简要概述了主要内容和结果。1.内模型和初等嵌入的极大性性质。例如，我们研究了在宇宙中，对于核模型K中的给定基数，Jonsson性质在什么情况下成立V.2.无限时间Turning机的研究。我们特别证明了，如果λ是可写序数的上确界，即在输入0上的计算的输出处出现的序数，并且γ是可钟序数的上确界，即在输入0上的停止计算的长度的序数，则λ=γ。3.证明了ZFC模型上的Cohen实数集要么为空要么为非空。4.连续统的基数不变量对不可数基数的组合性质的影响。例如，我们证明了零理想的可加性意味着马丁公理MA对任何科恩代数成立。另一方面，我们证明了连续统c是大的，零理想的覆盖以及组合原理 * 成立是相容的。5.称PSP（κ，Γ）成立，如果点类Γ中每个大小至少为κ的集合都有一个完美子集。我们展示了PSP（N D21 D2，G D2 N D2）成立当且仅当σ [大于或等于] N D22 N D2其中G D2 N D2是（至多）N D21 N D2多个开集的交的集合类，σ是控制数。自然数的置换群，其大小至少为最小非贫乏集实的基数。7.完成了由Shelah和Spinas开创的一个结果循环，我们得到了：如果c= N D 22 D 2，则在每一个可数无限域上都有一个Gross空间。