Mathematical Sciences: Topics in Logic and the Foundations of Mathematics

几个研究问题和问题领域是根据 调查，主要是在证明理论，与应用， 构造性和半构造性数学和理论 计算机科学 此外，还提出了一个新的问题领域， 与递归理论、逻辑和代数有关。 主题为 如下所示： 1. 代数与逻辑之间的递归论类比 (e.g., Higman有限嵌入定理和Kleene有限嵌入定理 公理化定理）。 2. 分析的子系统到弱的证明论约化 理论 (a)减少到PRA及以下。 (b)减少到PA。 3. 构造函数论的证明理论 班 (a)单调性一般原理的强度 归纳定义 (b)建设性编程系统的优势。 (c)多态 数理逻辑与理论计算机的相互作用 科学一直是，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.