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.
