Omega Model of Standard Calculus

本书中有两个发现和十七个成果。其中有十二个成果是关于当代数学的,另外的五个是解决了五个公元前250-550年之间的历史难题。发现之一是:本书系统地研究了不可分量(实数的空集合)及其性质。发现之二是:本书发现了标准微积分学的新模型-欧弥伽连续统模型。为了说清楚关于当代数学的十二个成果,令R代表实数集合且r∈R是任意的,这十二个成果是:Ⅰ.本书证明了实数集合不能填满一条建立了固定标架的欧几里德直线;Ⅱ.对确定了标架的Euclid直线L进行了完整的微分分拆,即L={-∞的右单子∪r的左单子∪r∪r的右单子∪∞的左单子}=欧弥伽连续统ΩΠ,并对无穷小量的积分建立了三条公理;Ⅲ.令ω代表r的左单子和r的右单子的共同测度,在标准数学中证明了ω是R之外的正无穷小;Ⅳ.对若当,卡拉特欧多里和勒贝格测度论中的两条公理给出了宇观的、宏观的和微观的反例,并给出了欧弥伽极限协调性测度的新概念;Ⅴ.由单个自然数的测度为零证明了自然数集合N的测度也等于零;并且由单个实数的测度为零证明了实数集合R的测度等于零;Ⅵ.在ΩΠ中定义了序和算术运算;Ⅶ.把外尔斯特拉斯极限改进为欧弥伽极限;Ⅷ.把狄特金分割改进为欧弥伽分割;Ⅸ.把康托连续统改进为欧弥伽连续统;Ⅹ.在ΩΠ中给出了欧弥伽定积分的定义;Ⅺ.

There are two discoveries and seventeen achievements in the book, where twelve achievements are of nowadays mathematics, and five achievements of difficult problems long standing for 2250 - 2500 years or more. The first discovery is to study systematically indivisibles (real number empty sets) in the book. The second discovery is to find a new model of standard calculus - Omega continuum model. In order to saying clearly the twelve achievements of nowadays mathematics, let R denote real number set and r ∈ R be arbitrary. The twelve achievements of nowadays mathematics are:Ⅰ . The book proves that the real number set can not fill up a Euclidean straight line with a fixed frame; Ⅱ . Gives a complete differential partition to a Euclidean straight line with a fixed frame L,i. e. L= { the right monad of -∞ U the left monad of r ∪ r ∪ the right monad of r ∪ the left monad of ∞} = Omega continuum fill, and sets up three axioms for the integral of infinitesimals; Ⅲ . Let to denote the common measure of the left monad and right monad of r, proves in standard mathematics that ω is a positive infinitesimal outside of R. ; Ⅳ. Gives cosmic, macro and micro counterexamples for two axioms in Jordan, Caratheodory, and Lebesgue measure theory, and sets up a new concept of Omega limit consistence measure; Ⅴ . Proves that the measure of natural number set N is equal to zero from a single natural number with zero measure; And proves that the measure of real number set R is equal to zero from a single real number with zero measure; Ⅵ - Defines order and arithmetic operators in ΩΠ; Ⅶ . Transforms Weierstrass limit into Omega limit; Ⅷ. Transforms Dedekind cut into Omega cut; Ⅸ Transforms Cantor continuum into Omega continuum; Ⅹ . Gives the definition of Omega definite integral in ΩΠ; Ⅺ. Gives three definitions for Omega definite integral for a real number function and three integrable function classes; Ⅻ . Accurately states and thoroughly proves the so - called Newton -Leibniz formula,and transforms it into Omega formula. The five achievements of ancient mathematics are: Ⅰ . The book rigorously proves the existence and systematically discusses the function of indivisibles, and develops the Pythagorean dictum 'All is number' to a more complete dictum 'All consists of indivisible, except for number'; Ⅱ . Solves the general dictum of Zeno of Elea;Ⅲ . Proves the hypothesis of Zhuang Zhou titled 'No thick may not be integrated. ' ; Ⅳ . Proves the hypothesis of Aristotle titled ' Aristotle denied that number can produce a continuum, inasmuch as there is no contact in numbers. '; Ⅴ. Proves the hypothesis of Zhuang Zhou titled 'Infinite times but not exhausted. ' Refer to [26], the author had a short Communication titled 'Standard Infinitesimal Calculus',in the International Congress of Mathematicians, Beijing 2002, August 20-28. The contents of the abstract of the short Communication is: 'The paper gives a complete differential partition to a Euclidean straight line with a fixed frame, and three axioms for the integral of infinitesimals indexed by real numbers; proves in standard mathematics there are positive infinitesimals outside of real number set; Gives cosmic, macro and micro counterexamples to two axioms in Jordan, Caratheodory, and Lebesgue measure theory; transforms Weierstrass limit into Huang limit, Cantor continuum into Huang continuum, and Newton - Leibniz formula into Huang formula.' Obviously the book is a rewriting of the short Communication.