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
“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。
袁萌
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
2 Construction of
the Hyperreals
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
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