层次理论，第1-3部分
Level Theory, parts 1-3
摘要：
本文件包括层次理论，第1-3部分。部分1.下面这个简单的故事介绍了纯集合的累积层次的概念：“集合是分阶段排列的。每个集合都是在某个阶段被发现的。在任何阶段S：对于在S之前找到的任何集合，我们找到一个集合，其成员恰好是这些集合。我们在S中找不到其他的东西。“令人惊讶的是，这个故事已经保证了集合是有序排列的，足以满足准范畴性。我通过介绍层次理论来说明这一点，层次理论是对斯科特、蒙塔古、德里克和波特的集合论的简化。部分2.潜势论者认为集合的概念是模态的。使用紧张的语言作为启发，下面的简单故事介绍了集合的潜在层次结构的想法：“总是：对于任何存在的集合，都有一个集合，其成员正是这些集合;没有其他集合。“令人惊讶的是，这个故事已经保证了良好的基础和持久性。此外，如果我们假设时间是线性的，那么随之而来的模态集合论在定义上几乎等同于非模态集合论;具体地说，等同于第1部分中发展的层次论。部分3.根据集合的一个非常自然的概念，每一个集合都有一个绝对补集。普通的累积层次结构完全否定了这种想法。但我们可以纠正这一点，同时保留经典逻辑。事实上，我们可以发展一个由有序层次上的集合组成的布尔代数。我通过介绍布尔层次理论来说明这一点，它将普通的层次理论（来自第一部分）与托马斯·福斯特、阿朗佐·丘奇和乌尔斯·奥斯瓦尔德的思想融合在一起。BLT巧妙地实现了康威博弈和超现实数; BLT的自然扩张在定义上与ZF等价。
英文摘要：
This document comprises Level Theory, parts 1-3. PART 1. The following bare-bones story introduces the idea of a cumulative hierarchy of pure sets: 'Sets are arranged in stages. Every set is found at some stage. At any stage S: for any sets found before S, we find a set whose members are exactly those sets. We find nothing else at S.' Surprisingly, this story already guarantees that the sets are arranged in well-ordered levels, and suffices for quasi-categoricity. I show this by presenting Level Theory, a simplifiation of set theories due to Scott, Montague, Derrick, and Potter. PART 2. Potentialists think that the concept of set is importantly modal. Using tensed language as an heuristic, the following bare-bones story introduces the idea of a potential hierarchy of sets: 'Always: for any sets that existed, there is a set whose members are exactly those sets; there are no other sets.' Surprisingly, this story already guarantees well-foundedness and persistence. Moreover, if we assume that time is linear, the ensuing modal set theory is almost definitionally equivalent with non-modal set theories; specifically, with Level Theory, as developed in Part 1. PART 3. On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, we can develop a boolean algebra of sets arranged in well-ordered levels. I show this by presenting Boolean Level Theory, which fuses ordinary Level Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Church, and Urs Oswald. BLT neatly implement Conway's games and surreal numbers; and a natural extension of BLT is definitionally equivalent with ZF.
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