Graphs on Generalised Baire Spaces

建议项目的研究是在公理集合论的范围内进行的。这一理论通常被视为所有数学的基础，因为每个数学概念在结构上都可以用无限集合来表示。尽管世界的大小是有限的，但我们可以通过图灵发现的程序和数字计算的建模来感受到我们计数的无穷大的理论影响：虽然计算机是有限的，但关于其能力的理论最好是在无限的背景下进行。现代集合论的创始人康托尔试图通过首先建立简单描述的集合的结果，然后建立更复杂的集合的结果，等等来解决关于实数行的子集的疑难问题。这就建立了描述集合论的概念。在1910年的S和20年的S的“古典时期”，俄国(苏斯林、吕津)和法国(博雷尔、勒贝格)学派的分析家们集中精力将结果建立到他们可以描述的水平：“波雷尔”集或“解析”集。例如，对于平面上的这些集合，即使这些区域不是被简单的曲线包围的区域，也可以形成“面积”的概念。然而，问题停留在这个层面上。勒贝格定义了一个超越解析的“射影集合”的层次结构，但他对发现它们是否在这个意义上是“可测量的”感到绝望。现代集合论已经发现了经典分析家陷入困境的原因：需要超越泽梅洛-弗拉恩克尔(Zermelo-Fraenkel，1920年在《S》一书中提出的)的标准使用的公理或公设。要让这些投射集正常运行，需要在集合的宇宙中假设更强大的“无穷公理”(也称为“大基数”)。一个令人惊讶但意义重大的发展是使用无限长的两人完全信息游戏。假设这样的游戏起到了制胜策略的作用。玩家交替进行整数步，游戏的长度与N的长度相同。从技术上讲，这些游戏被称为“贝尔空间上的游戏”。我们的项目是将这些想法中的一些重新聚焦于当前兴起的一个新的感兴趣的领域：“广义贝尔空间”：我们看的不是类型N的序列，它可以被解释为实数的十进制扩展，而是更长的序列，它的类型是康托的一个大的不可数基数，它仍然大于自然数的大小。在这个意义上，相关的概念游戏也更长，可能会，也可能不会，受到与早期游戏相同类型的分析的影响。我们还不知道。最初的贝尔空间通常被认为是无理数(留下的许多有理数不计入面积、度量等概念)。因此，我们可以认为广义版本是对实数线在这个特定方向上的广义。我们为什么要担心这一点？研究这种更强大的公理的意义要广泛得多：对于一般的数学分析师来说，强大的公理会影响他们对实数线的看法，而这一点现在才开始受到重视。纯数学的几个领域可以说直接受到集合论公理的影响。从更广泛的角度来看，对“无限”和“集合”的本质的理解在哲学上和一般人类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合理论家，但更广泛地说，是对这些问题感兴趣的数理逻辑学家和数学哲学家。集合理论在国际上非常活跃，在美国、以色列、奥地利、法国和德国都有重要的研究小组。然而，在英国，先进的集合论在某种程度上是代表不足的，集中在布里斯托尔、UEA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。

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.