COOPERATIVE RESEARCH OF MATHEMATICAL LOGIC AND ITS RELATED FIELDS

该项目的主要目的是数理逻辑、证明论、递归论、集合论、模型论、非标准分析和理论计算机科学的研究人员从数理逻辑在这些领域的应用的角度进行合作研究。在这个学期，最重要的发展是由Hiroakira Ono领导的非标准逻辑研究。通过限制几个结构规则而获得的逻辑在日本已经被几位逻辑学家研究过。然而，在Girald提出的线性逻辑的刺激下，小野及其合作者对亚结构逻辑的研究又重新活跃起来。在这种情况下，小野提出了非标准逻辑国际会议的建议，并计划在日本召开第一次会议。去年秋天在金泽举行的NSL'94会议上，宣布了许多关于非标准逻辑的重要结果。预计该分支将在未来得到更大的发展。该小组的理论家所领导的静雄卡莫制定了研究理想和组合性质的规定。他们在这个分支取得了几项突出的成果。以元桥为代表的模型论小组研究了稳定性理论，特别是稳定性理论在代数结构中的应用，得到了一些有趣的结果。由Teturou Kamae领导的非标准分析小组发展了泛函分析的非标准方法，并获得了几个重要结果。卡库达在几年前提出了无非标准模型的非标准分析方法，受此启发，Yasugi发展了具有无限长公式的语言元数学，并将其应用于非标准分析。鉴于研究成果对教育的减少，卡库达目前正在开发一种新的微积分教学方法，即利用谓词系统和编写一本微积分教科书。

The main aim of this project is that researchers in mathematical logic, proof theory, recursion theory, set theory, model theory, non-standard analysis and theoretical computer science cooperatively research in the view point of applications of mathematical logic to those fields. During the term, the most important development was the study of non-standard logic led by Hiroakira Ono. Logic obtained by restricting several structural rules had been studied by several logicians in Japan. Hoever, under the stimulus of emergence of linear logic by Girald, the study of substructural logic has been active again by Ono and cooperators. Under these circumstances, Ono proposed the international meeting of non-standard logic and planed to open the 1st meeting in Japan. The meeting, called NSL'94, was held in Kanazawa last fall, and many important results concerning non-standard logic were announced. It is highly expected this branch will be developed more in future. The group of set theorist led by Shizuo Kamo developed the study of ideals and combinatorial properties on sets. They obtained several outstanding results in this branch. The group of model theorist led by Motohashi studied the stability theory, especially applications of the theory to the algebraic structures, and obtained several intereting results. The group of non-standard anlysis led by Teturou Kamae developed the non-standard method of functional analysis, and obtained several important results. Kakuda proposed the method of non-standard analysis without non-standard models several years ago.Inspired by this method, Yasugi developed metamathematics of language with infinitely long formulas, and applied it to non-standard analysis.In view of the reduction of research results to education, Kakuda is now developping the new teaching method of calculus by using the system of predicates and writing a text book of calculus.