Inner Model Theory in Outer Models

外模型中的内模型理论

• 批准号: EP/J005630/1
• 负责人: Philip Welch
• 金额: $ 25.47万
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2012
• 资助国家: 英国
• 起止时间: 2012 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ005630%2F1
• 关键词: Inner; Model; Theory; Outer; Models

The research of the proposed project is within axiomatic set theory. This theory is usually seen as a basis for all of mathematics, since every mathematical concept can be expressed structurally in terms of sets. Much of what mathematicians do concerns infinite sets or collections and it is this notion of 'infinite' that set theorists try to elucidate. Although the world is of finite size, the theoretical effects of the infinity of counting numbers is felt through, eg, modelling of computation by programs and numbers as discovered by Turing: although computers are finite, theorizing about their capabilities is best done in an infinite context. In similar ways we model the finite world by using 'infinite structures' and theories.In set theory much fundamental work was done by Kurt Goedel in showing that certain axioms known as the Axiom of Choice and the property known as Cantor's Continuum Hypothesis (CH - that every set of numbers on the number line is either countable or of the same size as the whole line) were consistent with the universally accepted axiom set. He did this by developing a structure or 'inner model' of those axioms with those desired extra properties. This process of inner model building has come to be seen as fundamental to our understanding of the universe of all sets of mathematical discourse (known as 'V'). Goedel's structure, called 'L', is now widely generalised and strengthened to incorporate more and more potential properties, or stronger axioms, that may hold in V. This program of inner model analysis and building was initiated by Ronald Jensen in the 1970's, who discovered fundamental properties of 'L' (called its 'fine structure'). However Paul Cohen in 1962 showed by a radically new method called 'forcing' that this could not be the whole story: one could build syntactic or 'virtual' models of the axioms in which properties such as the CH failed: such properties we call independent of the axioms.The research being undertaken here is very novel in that it tries to ascertain to what degree the fine structure of inner models can hold in certain of these 'virtual' (which we call 'outer' in the project) models. These outer models are often built assuming strong axioms hold in V, and theorists using the forcing techniques try and preserve these axioms when building them. But is it possible to have such strong axioms with at the same time fine structure of an inner model? Or are they incompatible? This is broadly the question that this project wishes to investigate.Why should we be concerned about this? From the viewpoint of set theorists this is important as the program building inner models has run into difficulties, and model building is (perhaps temporarily) halted. We might ask: are there then mathematical reasons for this? This project can help elucidate fundamental incompatibilities (if any) between fine structure and strong axioms. But the implication of studying such stronger axioms are much wider: for the general mathematical analysts strong axioms affect how they view the real number line, and this is only now starting to be appreciated. Several areas of pure mathematics can be said to be directly affected by set theoretic axiomatics.In the wider perspective an understanding of the nature of 'infinity' and 'set' is of interest both philosophically and for the general human endeavour. We thus think of the beneficiaries of this research as principally set theorists, but more widely, mathematical logicians and philosophers of mathematics who are interested in these questions. Set Theory is very active internationally, with significant research groups in, eg, USA, Israel, Austria, France, Germany. The area has been recognised with a large European Research Grant called INFTY. However,in the UK advanced set theory is somewhat underrepresented, and is concentrated in Bristol and at UEA. This project will thus enhance the UK's standing and expertise in set theory.

拟议项目的研究是在公理集理论。这个理论通常被视为所有数学的基础，因为每个数学概念都可以用集合来表达。数学家所做的大部分工作都与无限集合或集合有关，集合理论家试图阐明的正是“无限”的概念。虽然世界的大小是有限的，但计数的无限性的理论效果是通过图灵发现的程序和数字的计算模型来感受的：虽然计算机是有限的，但关于它们的能力的理论最好是在无限的背景下进行的。在集合论中，库尔特·哥德尔（Kurt Goedel）做了很多基础性的工作，证明了被称为选择公理（Axiom of Choice）的某些公理和被称为康托连续统假设（Cantor's Continuum Hypothesis）的属性（CH --在数线上的每一组数字都是可数的或与整条线的大小相同）与普遍接受的公理集是一致的。他这样做是通过开发一个结构或“内部模型”的那些公理与那些所需的额外性质。这种内在模型的建立过程被视为我们理解所有数学话语集合（称为“V”）的基础。哥德尔的结构，被称为“L”，现在被广泛推广和加强，以纳入越来越多的潜在性质，或更强的公理，这可能在V.这个程序的内部模型分析和建设发起的罗纳德詹森在20世纪70年代，谁发现的基本性质“L”（称为其“精细结构”）。然而，保罗·科恩在1962年通过一种全新的方法“强迫”证明了这不可能是全部：人们可以建立一些公理的句法或“虚拟”模型，在这些公理中，CH等属性失败了：这些性质我们称之为独立于公理。这里进行的研究非常新颖，因为它试图确定内部模型的精细结构在某种程度上可以保持这些“虚拟”（我们在项目中称之为“外部”）模型。这些外部模型通常是在假设V中有强公理的情况下建立的，而使用强迫技术的理论家在建立它们时试图保持这些公理。但是，有没有可能在内部模型的精细结构的同时拥有如此强的公理呢？或者它们是不相容的？这也是这个项目想要研究的问题，我们为什么要关注这个问题呢？从集合论的观点来看，这是很重要的，因为程序构建内部模型遇到了困难，模型构建（也许暂时）停止了。我们可能会问：这有数学上的原因吗？这个项目可以帮助阐明精细结构和强公理之间的基本不相容性（如果有的话）。但是，研究这些更强公理的意义要广泛得多：对于一般的数学分析家来说，强公理会影响他们如何看待真实的数线，而这一点直到现在才开始得到重视。一些领域的纯数学可以说是直接影响到集理论公理。在更广泛的角度理解的性质'无限'和'集'是感兴趣的两个专业和一般人类的努力。因此，我们认为这项研究的受益者主要是集合理论家，但更广泛地说，是对这些问题感兴趣的数学逻辑学家和数学哲学家。集合论在国际上非常活跃，在美国、以色列、奥地利、法国、德国都有重要的研究小组。该地区已获得一项名为INFTY的大型欧洲研究资助。然而，在英国先进的集合论是有点代表性不足，并集中在布里斯托和东英吉利大学。因此，该项目将提高英国在集合论方面的地位和专业知识。

项目成果

Simplest possible locally definable well-orders

最简单的局部可定义良序

• DOI: 10.4064/fm281-7-2016
• 发表时间: 2017
• 期刊: Fundamenta Mathematicae
• 影响因子: 0.6
• 作者: Lücke P

LOCAL CLUB CONDENSATION AND L-LIKENESS

当地俱乐部的凝结和l-likeness

• DOI: 10.1017/jsl.2015.6
• 发表时间: 2015-12
• 期刊: Journal of Symbolic Logic
• 影响因子: 0.6
• 作者: Holy Peter;Welch Philip;Wu Liuzhen
• 通讯作者: Wu Liuzhen

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

大基数和 LIGHTFACE 可定义的良序，无需 GCH

• DOI: 10.1017/jsl.2013.41
• 发表时间: 2015
• 期刊: The Journal of Symbolic Logic
• 作者: FRIEDMAN S

PFA and Class Forcing

PFA 和类别强制

• 期刊: Mathematical Logic Quarterly
• 影响因子: 0.3
• 作者: Holy, P

Locally S 1 -definable well-orders of H(? + )

局部 S 1 - H(? ) 的可定义良阶

• DOI: 10.4064/fm226-3-2
• 发表时间: 2014
• 期刊: Fundamenta Mathematicae
• 影响因子: 0.6
• 作者: Holy P