Mathematical Logic

我们组织了五个工作组。这些组织相互交换了信息，并举行了几次会议。在这些会议上，研究人员讨论并公布了他们的研究结果。在下面，我们列出了各组的主要结果。第一组（证明论）：解释了弱归纳定义算术子系统与序数之间的关系。得到了几个关于反射原理和Paris-Harrington原理的结果。提出了基本序列的广义组合系统。第二组（集合论和模型论）：提出了具有模态的集合论和具有密集性的集合论。假设存在几个大的基数，构造了有趣的阿贝尔群。解释了非稳定环的几个性质。第三组（构造数学）：建立了一个模糊可计算理论。给出了Beeson系统PRS的一致性证明。提出了超有限机构的两层理论。第四组（逻辑结构理论）：证明了中间谓词逻辑中的模态运算符可以被量词解释。统一讨论了不含部分结构规则的逻辑的语法与语义的关系、微积分和范畴语法。第五组（非标准宇宙理论）：在非有限逻辑下，得到了无穷小微积分的一个解释。解释了迭代多项式在非标准分析中的几个性质。对于Hilbert不可约定理，给出了Z-If有限的一个条件，并证明了其界是由多项式给出的。提出了一种非标准集合理论，其中包括NST、IST、NSTE、NS2和*NST五种非标准集合理论的模型。

We organized five working groups. the groups exchanged informations each other, and opened meetings on several occasions. In those meetings, researchers discussed and announced results of their researches.In the followings, we list up the main results of the groups. Group I(Proof theory): The relations between subsystems of arithmetic with weak inductive definitions and ordinals were explained. Several results concerned with refrection pronciples and Paris-Harrington principles were obtained. Generalized built-up systems of fundamental sequences were proposed.Group II(Set theory and model theory): A set theory with modality and a set theory with intensionality were proposed. Assuming existence of several large cardinals, interesting abelian groups were constructed. Several properties of -stable rings were explained.Group III(Constructive mathematics): A theory of Fuzzy computability was made up. A consistency proof of Beeson's system PRS was given. A two-storied theory of transfinite mechanisms was proposed.Group IV(Theory of logical structures): It was shown that modal operators are interpretable by quantifiers in intermediate predlcate logics. The relation between the syntax and the semantics of logics without a part of structural rules, -calculus, and categorical grammars were uniformly discussed.Group V(Theory of non-standard universes): In a nonfinitary logic, an interpretation of infinitesimal caculus was obtained. Several properties of iterated polynomials in nonstandard analysis were explained. For Hilbert irreducibility theorem, a condition for that Z-If is finite was given, and it was shown that the bound is given by a polynomial. A nonstandard set theory, in which there are models of the five nonstandard set theories NST, IST, NSTE, NS2 and *NST, respectively, was proposed.