Mathematical Sciences: Topics in Logic and the Foundations of Mathematics

几个研究问题和问题领域正在调查中，主要是在证明论方面，应用于建构性和半建构性数学和理论计算机科学。此外，还提出了一个新的问题领域，将递归理论、逻辑和代数联系起来。主要内容如下：1.代数与逻辑之间的递归论类比(如希格曼有限嵌入定理和Kleene有限公理化定理)。2.分析子系统的证明论归约为弱理论。(A)降至PRA及以下。(B)减至PA。3.函数和类的建构性理论的证明论。(A)单调归纳定义的一般原则的力度。(B)建设性方案拟订系统的力量。(C)多态。数理逻辑和理论计算机科学之间的相互作用一直是并将继续是非常富有成果的。

Several research problems and problem areas are under investigation, primarily in proof theory, with applications to constructive and semi-constructive mathematics and theoretical computer science. In addition a new problem area is proposed, relating recursion theory, logic, and algebra. The topics are as follows: 1. Recursion-theoretic analogues between algebra and logic (e.g., Higman's finite embedding theorem and Kleene's finite axiomatizability theorem). 2. Proof-theoretic reduction of subsystems of analysis to weak theories. (a) Reduction to PRA and below. (b) Reduction to PA. 3. Proof-theory of constructive theories of functions and classes. (a) The strength of the general principle of monotone inductive definition. (b) The strength of a constructive programming system. (c) Polymorphism. The interplay between mathematical logic and theoretical computer science has always been and continues to be very fruitful.