A new foundation for mathematics. Naïve set theory in HYPE

集合论是数学中研究集合的理论，被广泛认为是所有数学知识的基础框架。然而，该理论的原始表述，尽管具有强大的哲学依据，但在哲学和数学本质上都容易受到悖论的影响，这可以追溯到直觉上被证明的数学实体以恶性循环的方式运行。避免悖论的标准方法是基于一个禁止循环的理论：然而，在许多数学设置中，这种循环似乎是可取的，并且是语言的内在特征。这个项目的目的是建立一个理论，恢复失去的循环和哲学直观性，同时没有悖论。要做到这一点，我们需要修改作为理论基础的推理系统，并采用所谓的非经典逻辑。我们选择研究具有强条件的非经典逻辑，其优点是在处理悖论时相对灵活，同时保持一定的数学强度。这种逻辑的一个例子是由Hannes Leitgeb在2019年开发的HYPE。HYPE表现为一种高内涵逻辑，即一种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。使用HYPE来发展集合论很有趣，原因有二：首先，它允许我们以一致的方式带回原始的或naïve集合概念，它将集合视为概念的扩展。其次，虽然它有一个相对良好的结果关系，但它有一个非常灵活的语义，这使得我们可以很容易地对循环实体建模。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理论来寻找新的，数学上可行的集合理论悖论的解决方案。该项目的重点将主要是语义性质：我们将基于模型研究不同的理论，以研究它们如何对循环实体建模，从基于HYPE的集合理论的语义处理开始。项目的目的是寻找一个在非经典性和强度之间有良好平衡的理论，并研究在集合论中使用具有强条件的逻辑的优势。这项研究采用严格的数学技术进行，将阐明数学哲学中长期存在的深刻问题，如圆的地位，如果成功，将提供一个数学家，计算机科学家和基础科学家感兴趣的新框架。

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.