歐氏幾何與非歐幾何的一致性研究

Research about the Consistency of Euclidean Geometry and Non-euclidean Geometry

王德全(TE-CHUAN WANG)

指導教授 ： 蔡行健

國立中正大學/文學院/哲學系/碩士(2014年)

摘要

本論文的研究主要工作分成三個部份：

一、以初階邏輯語言，重述歐基里德幾何公設系統（在本論文中特指希爾伯特所建立的公理系統）。為完成此項工作，在第一章中整理、補充並重述了希爾伯特的五組公理系統、定義、定理，同時在第二章中介紹了相關的初階邏輯與模型論的基本概念。

二、透過建立不同的模型來說明各公理間的相容性與獨立性。這些模型大多都是在歐氏幾何中建構，其建構的過程是有趣的，但有些模型在理解上，可能會用到較複雜的數學。

三、透過在歐氏幾何空間中建立非歐幾何的模型的方式，以此重新驗證歐氏幾何的一致性蘊涵非歐幾何的一致性。

關鍵字

一致性 ； 幾何 ； 邏輯 ； 模型

並列摘要

The main work of this thesis research is divided into three parts:

1. I restate Euclidean geometry postulate system in first–order logic language. To accomplish this work, I sort out the information including Hilbert's axioms, geometry definitions, and theorems in the first chapter, and I introduce the basic concepts of logic and model theory in chapter 2.

2. Relying on the establishment of different models, we can illustrate the compatibility and the independence between these axioms. Most of these models are constructed in Euclidean geometry, and the construction process of these models is interesting. But some models require more mathematical knowledge to realize their meaning.

3. According to the establishment of non–Euclidean geometry models in Euclidean space, we can show that the consistency of Euclidean geometry implies the consistency of non–Euclidean geometry.

並列關鍵字

consistency ； geometry ； logic ； model

參考文獻

1. Hilbert, David, Foundations of Geometry, The Open Court Publishing Company (2nd ed., 1971).

2. Enderton, Herbert B. , A Mathematical Introduction to Logic, A Harcourt Science and Technology Company (2nd ed., 1972).

3. 趙文敏，幾何學概論，九章，1992年7月二版。

4. Hodges, Wilfrid, A Shorter Model Theory, Cambridge University Press (1997).