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哥德尔预言无穷小微积分是未来的数学分析

(2019-09-18 11:36:10)

 

哥德尔预言无穷小微积分是未来的数学分析

 

    二十世纪世界伟大的数学家哥德尔预言非标准分析是未来的数学分析。

 

   哥德尔1974年预言的原文如下:

 

There are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” [33]. Kurt G¨odel, 1974.请见本文附件1

 

   注:本文附件2是发表于201328日的非标准分析论文,此文附有44篇珍贵的非标准分析论文。

 

袁萌  陈启清  917

 

附件1

 

[33] T. Runge. Hypernite probability theory and stochastic analysis within Edward Nelsons internal set theory. 2011. URL http://www10.informatik. uni-erlangen.de/Publications/Theses/2010/Runge_DA10.pdf.

 

附件2

 

Eoghan Staunton

 

ID Number: 09370803

 

Final Year Project

 

National University of Ireland, Galway

 

Supervisor: Dr. Ray Ryan

 

February 8, 2013

 

I hereby certify that this material, which I now submit for assessment on the programme of study leading to the award of degree is entirely my own work and has not been taken from the work of others save and to the extent that such work has been cited and acknowledged within the text of my work.

 

Author:

 

Eoghan Staunton

 

ID No:

 

09370803

 

Contents

 

1 Introduction   1

 

2 Construction of the Hyperreals  2

 

2.1 Our aim . . . . . . 2

 

2.2 Z, Q and R from N . . . . . . . . 2

 

.3 Free Ultralters . . . . .. . 4

 

2.4 Generating elements of R . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Arithmetic operations and inequalities in R . . . . . . . . . . . . . 7 2.6 Some Notation & Denitions . . . . . . . . . . . . . . . . . . . . . 9 2.7 Other Ultrapower Constructions . . . . . . . . . . . . . . . . . . . 9 2.8 The -transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.9Internal vs. External constants . . . . . . . . . . . . . . . . . . . . 11 2.10 Innitesimals and Hyperlarge numbers in R . . . . . . . . . . . . 11

 

3 The Transfer Principle 14 3.1 History and Importance . . . . . . . . . . . . . . . . . . . . . . . . 14

 

3.2 Mathematical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 L o´s’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4 The Transfer Principle . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

 

3.6 Nonstandard Analysis as a Tool in Classical Mathematics . . . . . 23

 

4 The History of Innitesimals 25 4.1 Use in Ancient Greek Mathematics . . . . . . . . . . . . . . . . . . 25 4.2 Geometers of the 17th century and Indivisibles . . . . . . . . . . . 26

 

4.3 The Development of Calculus . . . . . . . . . . . . . . . . . . . . . 26

 

4.4 Modern Nonstandard Analysis . . . . . . . . . . . . .. . 29

 

5 Applications of Nonstandard Analysis 29 5.1 Economics and Finance . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2

 

Selected Other Applications . . . . . . . . . . . . . . . . . . . . . . 33

 

6 Appraisal and Conclusion    34

 

1 Introduction

 

There are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future” [33]. Kurt G¨odel, 1974.

An innitesimal is a number that is smaller in magnitude than every positive real number. The word innitesimal comes from the Latin word innitesimus and was coined by the German mathematician Gottfried Wilhelm Leibniz around 1710 [1]. We learn early on in our study of standard analysis that nonzero innitesimals cannot exist. It is also true however that many people use the intuitive notion when trying to understand basic concepts in a

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