Mathematical Sciences: Some Recursion Theoretic Problems

DMS-96-22290 哈灵顿大学加州伯克利分校 本项目研究整数的递归可重集，其研究方式与整数的枚举无关，也与整数如何被视为编码信息无关。它是研究纯粹相互关系的集合。这项研究的主要成分是：集合的自同构-存在于集合之间的对称性;不变集-防止对称性的墙。这两种成分自然是相辅相成的。 该字段中使用的主要方法称为“优先参数”。这是一种用于执行可计算构造的技术，以确保适当的不可计算属性。当前的一个指导问题是：如果不变集具有非平凡的最小信息量，那么它的每个成员都必须具有最大信息量。众所周知，解决这个问题将涉及到当前自同构技术的扩展。 一个集合是“re”（递归可重复的），如果它可以被可计算地列出。例如，所有可能的数学定理（从通常的公理推导出来）的集合是re。这个项目研究的是它们自己。彭罗斯在他的书《皇帝的新思维》中提出了计算和意识之间的类比。这个项目寻求的可能性，在更紧张的类比之间的重置（独立于他们的枚举）和意识的永恒方面。现有的一些成果可被视为这一方向的指示。这个字段中使用的主要方法称为“优先参数”。这是一种通过建立一个广泛的“世界观”领域来实现共同目标的方法，所有人都以各自不同的方式朝着共同目标努力，其中大多数是相互冲突的。该方法的一个有趣的特点是，必须允许所有合理的世界观，包括那些否认共同目的的世界观（这些世界观通过朝着自己的方向努力来完成自己的工作）。这个项目的成功将加强将意识与优先论证类宇宙联系起来的类比的潜在深远影响。

DMS-96-22290 Harrington, U. Cal-Berkeley This project studies the recursively enumerable (re) sets of integers, in a way which is independent of their enumeration and independent of how integers are viewed as coding information. It is the study of re sets purely in relation to each other. The main ingredients of this study are: automorphisms of the collection of re sets - the symmetries that exist among re sets; and invariant sets - the walls that prevent symmetries. These two ingredients naturally go hand-in-hand. The main method used in this field is called `priority arguments'. This is a technique for performing computable constructions that ensure appropriate non-computable properties. A current guiding problem is: if an invariant set has non-trivial minimum information content then each of its members must have maximum information content. It is known that settling this would involve an extension of current automorphism techniques. A set is "re" (recursively enumerable) if it can be computably listed. For example, the set of all possible theorems of mathematics (deduced from the usual axioms) is re. This project studies the re sets themselves. Penrose, in his book The Emperor's New Mind, suggests an analogy between computing and consciousness. This projects seeks possibilities in the more strained analogy between re sets (independent of their enumeration) and the timeless aspects of consciousness. There are existing results which can be seen as indicative of this direction. The main method used in this field is called "priority arguments". This is a method whereby a common purpose can be achieved by establishing an extensive field of "world views", all working towards the common goal in their own separate ways, most of which are conflicting. One interesting feature of the method is that all reasonable world views must be allowed, including those that deny the common purpose (those world views do their jobs by working towards their own ne gations). Success in this project would strengthen the potentially profound implications of the analogy linking consciousness with a priority argument -like universe.