Self-similarity and the Universal Steiner Triple System

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.