Marriage of non-standard analysis and computability theory toward the light of algorithmic randomness

本研究的主要目的是为抽象集合论传统上发展起来的非标准论证奠定逻辑基础。基于非标准方法，在二阶算术的可计算弱子系统上，得到命题的构造性内容。为了为二阶算法提供非标准方法，我们需要研究它们的非标准模型，这些模型实际上是实数（无限序列）的集合。在实数集合中，许多包含许多随机元素，这也是我们在本研究中探索的。

The main purpose of this research is to lay a logical foundation for non-standard arguments, which have been developed in abstract set theory traditionally. Basing non-standard methods on weak subsystems of second order arithmetic connected with computability, we can obtain constructive contents of propositions by such methods. To device non-standard methods for second order arithmetic, we need to investigate their non-standard models, which are indeed sets of reals (infinite sequences). Amongst the sets of reals, many contain many random elements, which we also explore in this study.