Self-similarity and the Universal Steiner Triple System

在这个项目中，我们将看到一个特别有趣的无限结构。系统具有与原系统相同（同构）的适当子系统。因此，在结构内部，有一个自身的副本。此外，它包含无限多个非嵌套的“副本”（它们实际上是不相交的，除了一个三元组）。更有趣的是，当你把两个这样的“副本”放在原始结构中时，我们会得到一个直观上比原始结构更复杂的新结构（与原始结构不同构）。结构似乎比本身更复杂，这似乎是一个悖论....但在无限世界中，这并不是一个悖论。这就是无限世界如此迷人的原因。该项目使用模型理论（数学逻辑的一个分支）的思想来回答与设计理论有关的问题。模型论关注的是你能用一种形式语言对对象说些什么。我们通常使用一种数学上足够表达的语言来捕捉语言中物体的要点，但这是尽可能简单的。利用模型理论的工具，我们将回答这个结构的自相似性问题。该结构是一个可数无限Steiner三重系统。给定一个由v个元素组成的集合X （v大于或等于3）和一个由X的3个子集（三元组）组成的集合B，使得X的每一个2个子集恰好出现在B的一个三元组中，则B称为斯坦纳三元组系统。有许多关于有限斯坦纳三重系统的数学。有趣的反常性质开始出现在无限的情况下，这就是我们将要研究的。

In this project we will look at a particularly interesting infinite structure. The system has proper subsystems which are identical (isomorphic) to the original system. So, inside the structure, there is a copy of itself. Furthermore it contains infinitely many of these 'copies' which are non-nested (they are in fact disjoint except for one triple). Even more interestingly when you put two of these 'copies' together inside the original structure, we make a new structure that is intuitively is more complex than the original (not isomorphic to the original). The structure seems somehow more complex than itself which seems like a paradox .... but it isn't a paradox in the infinite world. This is what makes the infinite world so fascinating.The project uses ideas from Model Theory (a branch of mathematical logic) to answer questions relating to Design Theory. Model Theory is concerned with what you can say about objects in a formal language. We usually use a language which is mathematically expressive enough to capture the essential points about objects in the language but that is as simple as possible. With the tools of Model Theory we will answer questions of self-similarity for this structure. The structure is a countably infinite Steiner triple system. Given a set, X, of v elements (v is greater than or equal to 3) together with a set B of 3-subset (triples) of X such that every 2-subset of X occurs in exactly one triple of B. Then B is called a Steiner triple system.There is much mathematics concerning finite Steiner triple systems. Interesting anomalous properties start to occur for the infinite cases, and it is these that we will investigate.