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# Takeshi YAMAZAKI【山崎 武】, Assoc. Professor

## Research Field

Logic and Foundations of Mathematics

## Research Interests

The formal system in which to study the natural numbers, 0, 1, 2, 3, …, is called the arithmetic of the 1st order, and another to study both the natural numbers and the set of natural numbers is called the arithmetic of the 2nd order. One characteristic of the arithmetic of the 2nd order is that, in addition to that there are the sequence, sum, and product operations and mathematical induction for natural numbers, it has the Set Existence Axiom that guarantees that you can have a "set" for any collection of natural numbers that you can define (using the predetermined language).

It has been known from the past that many parts of mathematics that you studied at the undergraduate course of university are to progress in the framework of arithmetic of the 2nd order. For that, the set existence axiom plays an important role. One research subject is to find a very limited form of the set existence axiom. Speaking in greater detail, I am researching the following two subjects mainly.

(1) To classify mathematical theorems using the partial systems that are characterized by the set existence axiom (such field of study is called "reverse mathematics", by the way), and

(2) To consider the partial set class of natural numbers, and associated partial systems and models.

Also studied are analyses of the random nature from the recursive theory perspective, the degree structure of the sets residing in the Baire space and the Cantor space, and reverse mathematics with respect to the arithmetic of the 3rd order.

## What is Expected from Students

Following are what I expect from you.

(1) Get involved in seminars and enjoy the study and research for yourself

(2) Be sincere to the mathematical fields and theories that you study. We welcome you when you come and see us for questions to ask in self-seminars if you like or to attend the seminar that is currently going on for your reference. The topics of the current seminar (of 2008), including those of the other self-seminars continuing separately, are (i) Reverse mathematics and recursive theory with respect to the combinatorial theory, (ii) Research on the degree structure with respect to the recursive and closed set in the Cantor space, and (iii) Descriptive set theory.

## Master's Thesis Supervision

- "Expansion of Algebra with regard to the Arithmetic of the 2nd Order"
- "Linear Theory and Functions Representation"

## Remarks

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