A new foundation for mathematics. Naïve set theory in HYPE

数学的新基础。

• 批准号: 2732306
• 金额: --
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2732306
• 关键词: new; foundation; mathematics.; Na; ve

Set theory, the theory studying collections in mathematics, is widely regarded to be the foundational framework for all of mathematical knowledge. However, the original formulation of the theory, despite having a strong philosophical justification, was susceptible to paradoxes, both philosophical and mathematical in nature, which can be traced back to the presence of intuitively justified mathematical entities behaving in a viciously circular way. A standard way to avoid the paradoxes is based on a theory which bans circularity: however, in many mathematical settings, this very circularity seems desirable, and an intrinsic feature of languages. The aim of this project is to build a theory which restores the lost circularity and philosophical intuitiveness, while being paradox-free. To do so, we will need to modify the system of reasoning which underlies the theory, and adopt a so-called non-classical logic. We choose to study non-classical logics with a strong conditional, which have the advantage of being relatively flexible to deal with paradoxes, while maintaining a certain mathematical strength. One example of such logic is HYPE, developed by Hannes Leitgeb in 2019. HYPE is presented as a hyperintensional logic, i.e. a system of reasoning which is suitable to deal with fine-grained logical contexts, such as contexts which deal with properties or belief. Using HYPE to develop a set theory is interesting for two reasons: firstly, it allows us to bring back the original, or naïve notion of set, which views collections as extensions of concepts, in a consistent way. Secondly, while having a relatively well-behaved consequence relation, it has a very flexible semantics, which allows us to model circular entities easily. The project will be devoted to finding new, mathematically viable solutions to the paradoxes of set theory by finding new alternative set theories with a sufficient mathematical content. The focus of the project will be mostly semantical in nature: we will study different theories based on their models, to investigate how they model circular entities, starting from a semantical treatment of a set theory based on HYPE. The aim of the project is to find a theory with a good balance between non-classicality and strength, and to investigate the advantages of employing logics with strong conditionals in set theory. This study, conducted with rigorous mathematical techniques, will shed light on longstanding and deep questions in the philosophy of mathematics, such as the status of circularity, and, if successful, will deliver a new framework which will be of interest to mathematicians, computer scientists and foundationally-minded scientists.

集合论是研究数学集合的理论，被广泛认为是所有数学知识的基础框架。然而，该理论的最初表述尽管具有很强的哲学依据，但在哲学和数学上都容易受到悖论的影响。本质上，这可以追溯到直觉上合理的实体以恶性数学循环方式表现的存在，避免悖论的标准方法是基于禁止循环的理论：然而，在许多数学设置中，这种循环似乎是可取的。 ，以及一个内在的该项目的目的是建立一种恢复失去的循环性和哲学直观性的理论，同时不存在悖论。为此，我们需要修改该理论的推理系统，并采用一种新的理论。所谓非经典逻辑，我们选择研究相对条件性较强的非经典逻辑，其优点是能够灵活处理悖论，同时保持一定的数学实力，这种逻辑的一个例子就是HYPE。通过汉内斯Leitgeb，2019 年。HYPE 被认为是一种超内涵逻辑，即适合处理细粒度逻辑上下文的推理系统，例如处理属性或信念的上下文，这对于两个人来说都很有趣。原因：首先，它使我们能够以一致的方式带回原始的或幼稚的集合概念，即集合作为概念的延伸。良好的结果关系，它具有非常灵活的语义，这使我们能够轻松地对循环实体进行建模该项目将致力于通过寻找具有足够数学能力的新的替代集合论来寻找新的、数学上可行的解决方案来解决集合论的悖论。该项目的重点本质上主要是语义：我们将基于其模型研究不同的理论，从基于 HYPE 的集合理论的语义处理开始研究它们如何建模循环实体。就是找到一个具有良好平衡性的理论这项研究采用严格的数学技术进行，将揭示数学哲学中长期存在和深刻的问题，例如数学的地位。循环性，如果成功的话，将提供一个新的框架，数学家、计算机科学家和有基础的科学家都会对此感兴趣。