Graphs on Generalised Baire Spaces

广义贝尔空间上的图

基本信息

  • 批准号:
    EP/V009001/1
  • 负责人:
  • 金额:
    $ 38.72万
  • 依托单位:
  • 依托单位国家:
    英国
  • 项目类别:
    Research Grant
  • 财政年份:
    2021
  • 资助国家:
    英国
  • 起止时间:
    2021 至 无数据
  • 项目状态:
    已结题

项目摘要

The research of the proposed project is within axiomatic set theory. This theory is usually seen as a foundation for all of mathematics, since every mathematical concept can be expressed structurally in terms of infinite sets. Although the world is of finite size, the theoretical effects of the infinity of our counting numbers, ``N'', 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.G. Cantor, the originator of modern set theory, tried to solve knotty problems about subsets of the real number line by establishing results first for simply described sets, then building up for more complicated ones, etc. This founded the concept of 'descriptive set theory'. In the `classical period' of the 1910's and 20's the Russian (Suslin, Luzin) and French (Borel, Lebesgue) schools of analysts worked intensively on establishing results up to the level they could describe: 'Borel' or 'analytic' sets. For example, for these sets in the plane an idea of "area'' can be developed even if these are not regions enclosed by a simple curve. However matters were stuck at this level. Lebesgue had defined a hierarchy of "projective sets'' beyond the analytic, but despaired of discovering whether they could be 'measurable' in this sense. Modern set theory has discovered why the classical analysts were stuck: axioms, or postulates, beyond the standardly used ones of Zermelo-Fraenkel (developed in the 1920's "ZF'') were needed. Either stronger "axioms of infinity" (also called "large cardinals'') were needed to be assumed in the universe of sets to get these projective sets to behave properly. One surprising but significant development was the use of infinitely long two person perfect information games. Assuming such games had winning strategies played a role. Players alternated integer moves, and the games had length the same type as N. These are technically known as "games on Baire space''. Our project is to refocus some of these ideas on a current new area of interest that has sprung up: "Generalised Baire spaces'': instead of sequences of type N that can be construed as a decimal expansion of a real number, we look at yet longer sequences the type of one of Cantor's large uncountable cardinal numbers, that is yet greater than the size of the natural numbers. The associated conceptual games are also longer in this sense, and may, or may not, be susceptible to the same kinds of analysis as the earlier ones. We do not yet know. The original Baire space is often identified with the irrational numbers (the countably many rationals left over not counting towards notions of area, measure etc.) We can thus think of the Generalised versions as generalising the real number line in this particular direction.Why should we be concerned about this? 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. However, in the UK advanced set theory is somewhat underrepresented, and is concentrated in Bristol, UEA and at Leeds. 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无限集合来表达。虽然世界的大小是有限的,但我们计数的数字“N”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。同样,我们用“无限结构”和理论来模拟有限世界。康托,创始人现代集合论,试图解决棘手的问题子集的真实的号码线建立的结果首先简单描述的一套,然后建立了更复杂的,等等,这奠定了概念的“描述集理论”。在20世纪10年代和20年代的“古典时期”,俄罗斯(Suslin,Luzin)和法国(Borel,Lebesgue)的分析师学校致力于建立他们可以描述的水平的结果:“Borel”或“分析”集。例如,对于平面中的这些集合,即使这些集合不是由简单曲线包围的区域,也可以发展出“面积”的概念。然而,事情就卡在了这个层面上。勒贝格定义了一个超越分析的“投射集”的层次,但对发现它们是否可以在这个意义上“测量”感到绝望。现代集合论已经发现了为什么经典分析师被卡住了:公理,或公设,超越了Zermelo-Fraenkel(在20世纪20年代“ZF”开发)的标准使用。要么更强的“无穷公理”(也称为“大基数”)需要在集合的宇宙中假设,以使这些投影集正确地表现。一个令人惊讶但重要的发展是使用无限长的两人完美信息博弈。假设这样的游戏有获胜的策略发挥了作用。玩家交替整数移动,游戏长度与N相同。这些在技术上被称为“Baire空间上的游戏”。我们的项目是将这些想法中的一些重新聚焦于当前出现的一个新的感兴趣的领域:“广义Baire空间”:而不是可以被解释为真实的数的十进制扩展的N型序列,我们研究更长的序列,康托的大不可数基数之一的类型,它大于自然数的大小。在这个意义上,相关的概念游戏也更长,并且可能会或可能不会受到与早期游戏相同的分析。我们还不知道。最初的Baire空间通常被认为是无理数(剩下的可数有理数不计入面积、测度等概念)。因此,我们可以把广义版本看作是在这个特定方向上对真实的数线的广义化,我们为什么要关心这个呢?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。纯数学的几个领域可以说是直接受到集合论公理的影响。从更广泛的角度来看,理解“无限”和“集合”的本质,无论是在理论上还是在人类的普遍努力中,都是有意义的。因此,我们认为这项研究的受益者主要是集理论家,但更广泛地说,数学逻辑学家和哲学家的数学谁是感兴趣的这些问题。集理论是非常活跃的国际上,与重要的研究小组,如美国,以色列,奥地利,法国,德国。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,东英吉利大学和利兹。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

期刊论文数量(10)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Canonical Truth
规范真理
  • DOI:
    10.1007/s10516-022-09631-5
  • 发表时间:
    2022
  • 期刊:
  • 影响因子:
    0.5
  • 作者:
    Carl M
  • 通讯作者:
    Carl M
Decision Times of Infinite Computations
无限计算的决策时间
Ideal topologies in higher descriptive set theory
更高描述集合论中的理想拓扑
Uniformization and Internal Absoluteness
统一化和内部绝对性
ASYMMETRIC CUT AND CHOOSE GAMES
不对称剪切和选择游戏
  • DOI:
    10.1017/bsl.2023.31
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    HENNEY-TURNER C
  • 通讯作者:
    HENNEY-TURNER C
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Philip Welch其他文献

Possible-Worlds Semantics for Modal Notions Conceived as Predicates
  • DOI:
    10.1023/a:1023080715357
  • 发表时间:
    2003-04-01
  • 期刊:
  • 影响因子:
    1.000
  • 作者:
    Volker Halbach;Hannes Leitgeb;Philip Welch
  • 通讯作者:
    Philip Welch
Richness and Reflection ∗
丰富性和反思*
  • DOI:
  • 发表时间:
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Neil Barton;Tim Button;Katherine Cuccuru;Marcus Gi;Leon Horsten;Peter Koellner;Penelope Maddy;Ian Rumfitt;Josephine Salverda;Zeynep Soysal;Sean Walsh;Philip Welch
  • 通讯作者:
    Philip Welch
Lower consistency bounds for mutual stationarity with divergent uncountable cofinalities
  • DOI:
    10.1007/s11856-018-1771-4
  • 发表时间:
    2018-09-08
  • 期刊:
  • 影响因子:
    0.800
  • 作者:
    Dominik Adolf;Sean Cox;Philip Welch
  • 通讯作者:
    Philip Welch
A Consideration of the Inheritance of Musical Talent on the Occasion of the Mozart Bicentenary
莫扎特二百周年之际对音乐人才传承的思考
  • DOI:
  • 发表时间:
    1993
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Philip Welch
  • 通讯作者:
    Philip Welch
The organic triangle cycle experience and status
  • DOI:
    10.1016/j.egypro.2017.09.238
  • 发表时间:
    2017-09-01
  • 期刊:
  • 影响因子:
  • 作者:
    Lance Hays;Philip Welch;Patrick Boyle
  • 通讯作者:
    Patrick Boyle

Philip Welch的其他文献

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{{ truncateString('Philip Welch', 18)}}的其他基金

Inner Model Theory in Outer Models
外模型中的内模型理论
  • 批准号:
    EP/J005630/1
  • 财政年份:
    2012
  • 资助金额:
    $ 38.72万
  • 项目类别:
    Research Grant
An analysis of Spector Classes associated with quasi-inductive definitions
与准归纳定义相关的 Spector 类的分析
  • 批准号:
    EP/G020841/1
  • 财政年份:
    2008
  • 资助金额:
    $ 38.72万
  • 项目类别:
    Research Grant

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伍德尔猜想和广义伯奇-富尔克森猜想的新视角
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    EP/X030989/1
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    2024
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    EP/W014882/2
  • 财政年份:
    2023
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    $ 38.72万
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    Research Grant
Asymptotics of Toeplitz determinants, soft Riemann-Hilbert problems and generalised Hilbert matrices (HilbertToeplitz)
Toeplitz 行列式的渐进性、软黎曼-希尔伯特问题和广义希尔伯特矩阵 (HilbertToeplitz)
  • 批准号:
    EP/X024555/1
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    2023
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Understanding cognitive and behavioural mechanisms of Generalised Anxiety Disorder in adolescents
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