Mathematical Sciences: Some Recursion Theoretic Problems

加州大学伯克利分校哈林顿分校这个项目研究整数的递归可枚举(Re)集，其方式独立于它们的枚举，也独立于整数被视为编码信息的方式。它纯粹是关于重集之间相互关系的研究。这项研究的主要内容是：re集集合的自同构--re集之间存在的对称性；以及不变集--阻止对称性的墙。这两种成分自然是相辅相成的。此字段中使用的主要方法称为‘优先级参数’。这是一种用于执行可计算构造以确保适当的不可计算属性的技术。当前的一个指导问题是：如果一个不变集具有非平凡的最小信息量，那么它的每个成员必然具有最大信息量。众所周知，解决这一问题将涉及当前自同构技术的扩展。如果一个集合是可计算列出的，那么它是“re”(递归可枚举的)。例如，所有可能的数学定理的集合(从通常的公理推导而来)是Re。这个项目研究的是RESET本身。彭罗斯在他的《皇帝的新思维》一书中提出了计算和意识之间的类比。这个项目在重新设定(独立于它们的列举)和意识的永恒方面之间更紧张的类比中寻找可能性。现有的结果可以被视为这一方向的指示性结果。此字段中使用的主要方法称为“优先级参数”。这是一种方法，可以通过建立广泛的“世界观”来实现共同目标，所有这些世界观都以各自不同的方式努力实现共同目标，其中大多数是相互冲突的。这种方法的一个有趣特征是，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(这些世界观通过努力实现自己的新目标来发挥作用)。这个项目的成功将加强将意识与优先争论一样的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.