## 概要

1. $\mathrm{ZF}$+$\mathrm{DC}$+「任意の実数の集合が可測」
2. $\mathrm{ZF}$+$\mathrm{CC}$+「任意の実数の集合が可測」
3. $\mathrm{ZFC}$ +「到達不能基数の存在 ($\mathrm{IC}$)」

ここで、到達不能基数とは、その存在からZFCの無矛盾性が証明されるため、$\mathrm{ZFC}$単体からは存在を証明出来ない巨大基数の一種で、その中でも最も大人しいものです。

(3)「到達不能基数の存在」から (1)「ZF+DC+任意の実数の集合が可測」 を導いたのがSolovayで、当初到達不能基数の仮定は落とせると予想していましたが、後にShelahが (1)または(2)から(3)を示し、必要十分であることが明らかにされました1

また、Solovayの体系は通常のフルパワーの選択公理を仮定した解析学の体系とは異なる理想的な性質が多く成り立つため、既存の解析学の基礎に対する代替的な体系たり得ます。 本稿では、その体系についても非常に簡単な分析を行います。

### Abstract

It is widely known that, under the full Axiom of Choice, there exists sets of reals without Baire property and which is not Lebesgue measurable. On th other hand, it is also known that it is enough, to develop the basic theory of analysis and measure theory, to assume the Axiom of Dependent Choice (DC) or the Axiom of Countable Choice (CC) which are weakened form of AC.

In this thesis, we will treat the following theorems due to Solovay and Shelah:

The following three systems are equiconsitent (i.e. have the same consistency strength):

1. $\mathrm{ZF}$ + $\mathrm{DC}$ + “Every set of reals is Lebesgue measurable”
2. $\mathrm{ZF}$ + $\mathrm{CC}$ + “Every set of reals is Lebesgue measurable”
3. $\mathrm{ZFC}$ +“The exisntence of an inaccessible cardinal ($\mathrm{IC}$)”

First, Solovay showed the direction (3) to (1) and he conjectured the use of inaccessibles can be dropped. But, later, Shelah showed the direction (1) or (2) to (3), and they turned out to be equiconsitent 2.

Furthermore, Khomskii recently generalized the result of Solovay to general notion of $I$-genericity, which subsumes the Baire Property and Lebesgue measurability. Actually, we will treat this generalized form of Solovay Theorem.

In addition, the system suggested by Solovay can be regarded as an alternative foundation to develop analysis. We also give really brief survey on this topic.

## Contents

### Reference Information

@mastersthesis{ISHII:2016sf,
Author = {Hiromi ISHII},
Institution = {Tsukuba University},
Title = {On Regularity Properties of Set of Reals and Inaccessible Cardinals},
School = {Graduate School of Pure and Applied Sciencies},
Month = 2,
Year = {2016}}

### History

• 2016/02/24 Published

1. また、Shelahの論文では「任意の実数の集合がBaireの性質を持つ」には到達不能基数が不要な事を示しましたが、本稿では扱いません。↩︎

2. Shelah also showed that “Every set of reals has Baire Property” doesn’t require any inaccessibles, but we don’t step in this direction in this thesis.↩︎