A new foundation for mathematics. Naïve set theory in HYPE

集合论是研究数学中集合的理论，被广泛认为是所有数学知识的基本框架。然而，这一理论的最初表述，尽管有很强的哲学合理性，但容易受到悖论的影响，无论是哲学上的还是数学上的，这些悖论可以追溯到直觉上合理的数学实体的存在，这些实体的行为是恶性循环的。避免悖论的一种标准方法是基于一种禁止循环的理论：然而，在许多数学环境中，这种循环似乎是可取的，也是语言的内在特征。这个项目的目的是建立一种理论，在没有悖论的同时，恢复失去的循环性和哲学直觉性。要做到这一点，我们需要修改作为该理论基础的推理系统，并采用所谓的非经典逻辑。我们选择研究条件较强的非经典逻辑，它的优点是相对灵活地处理悖论，同时保持一定的数学实力。这种逻辑的一个例子是Hannes Leitgeb在2019年开发的Hype。Hype被描述为一种超内涵逻辑，即一种适用于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。使用Hype来发展集合论很有趣，原因有两个：首先，它允许我们以一致的方式带回原始的或天真的集合概念，即将集合视为概念的扩展。其次，虽然它具有相对良好的因果关系，但它具有非常灵活的语义，这使得我们可以很容易地对圆形实体进行建模。该项目将致力于通过寻找具有足够数学内容的新的替代集合论，为集合论的悖论找到新的、在数学上可行的解决方案。该项目的重点将主要是语义的性质：我们将研究基于他们的模型的不同理论，以调查他们如何对循环实体建模，从基于炒作的集合理论的语义处理开始。该项目的目的是找到一种在非古典性和强度之间取得良好平衡的理论，并探讨在集合论中使用带有强条件句的逻辑的优势。这项用严谨的数学技术进行的研究，将阐明数学哲学中长期存在的深层次问题，如圆的地位，如果成功，将提供一个新的框架，将引起数学家、计算机科学家和具有基础思想的科学家的兴趣。

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.