Mathematical Sciences: Recursive Model Theory

奈特打算继续研究递归模型理论中的问题，并分析组织和思考嵌套优先参数的方法。现在有几种这样的方法：哈林顿开发的“工作者”方法，两个版本的“阿尔法系统”，都是由Ash开发的，以及Lemmp和Lerman的方法。Worker的方法和最初的阿尔法系统在递归模型理论中都产生了相当多的结果。奈特有一个关于工人的元定理，类似于阿什为他最初的阿尔法系统证明的那个定理，只是没有那么多假设是r.e.e，而且条件更复杂。奈特希望为工人简化和改进元定理。在奈特的元定理中，就像在Ash的原始元定理中一样，底部产生的对象是递归的。Ash的新的阿尔法系统(比旧的更简单)，以及勒姆普和勒曼的方法，都是为了产生一个r.e。物体在底部。递归模型理论(群的递归表示的存在性等)以及递归理论中存在许多问题，这些新方法应该在这些问题上产生结果。递归理论作为数学基础中的一个主题，其动机是将算法可计算的性质形式化。该算法不需要是一个实用的算法，但它必须存在并在一定的有限步数后原则上终止。

Knight intends to continue to work on problems in recursive model theory, and to analyze methods for organizing and thinking about nested priority arguments. There are now several such methods: the method of "workers", developed by Harrington, two versions of "alpha-systems", both developed by Ash, and a method of Lemmp and Lerman. The method of workers and the original alpha-systems have each yielded quite a number of results in recursive model theory. Knight has a metatheorem for workers, similar to the one Ash proved for his original alpha-systems, except that not so much is assumed to be r.e., and the conditions are more complicated. Knight hopes both to simplify and to improve the metatheorem for workers. In Knight's metatheorem, as in Ash's original metatheorem, the object produced on the bottom is recursive. Ash's new alpha-systems (simpler than the old), and the method of Lemmp and Lerman, are designed to produce an r.e. object on the bottom. There are many problems in recursive model theory (existence of recursive presentations of groups, etc.), as well as in recursion theory, on which these new methods ought to yield results. Recursion theory as a topic in the foundations of mathematics is motivated by a desire to formalize the property of being computable by an algorithm. The algorithm need not be a practical one, but it must exist and terminate in principal after some finite number of steps.