外的非标准分析学

The External Nonstandard Analysis

Ⅰ、本文论述了数系的扩大,引进了潜在数,如最小的正的潜无穷大p等,最后将超有理数集Q扩大为实序宇宙U_2。Ⅱ、本文介绍了外的非标准分析的理论基础,如逻辑扩大的转移原则,外的生成原则,比较原则和潜在数的表示式等。Ⅲ、本文证明了:1、标准实数集R的测度为0;2、每个潜在数的测度大于零;3、标准实数集的基数等于最小的正的潜无穷大p,同一序数p,等于两个不同的基数ω和c,故在序型U_2中连续统假设是一个不可判定的定理,这是本文对Hilbert第一问题的简明回答;4、对附有δ函数型无穷小扰动的Newton位势的解;得到了一些在标准分析中不能发现的函数项。Ⅳ、本文对一些历史上的数学问题作了回答,如:1、Pythagoras,Democritus和Plato先后提出了数学上的单子论或原子论,猜想数学中存在不可分的连续体,现在我们引入了潜在数,它们是一个个不可分的数学上的连续的实体,这样圆满地肯定了他们的猜想。2、对于公元前五世纪Zeno的总格言:“这种东西,加到别的东西上不使其增大,从别的东西中减去又不使其减小,不过是子虚乌有而已。”芝诺设置的这个总格言难住了人们两千四百多年,现在被我们从反面加以破解。3、对于Aristotle的猜想:“否认数能够产生一个连续统,因为数与数之间不能互相接触。”本文对上述猜想给以精确的陈?

1 、The paper discusses the enlargements of number systems, introduces potential numbers, for example, the least positive potential infinitelarge p and enlarges the hyper rational number set * Q to be a real ordered universe U2. Ⅱ、 The paper introduces the theoretical foundations of external nonstandard analysis, the transfer principles, external producing principles and comparative principles of logic enlargement, and the expression of any potential number. Ⅲ 、The paper proves: 1. The measure of the standard real number set R is equal to 0;2. The measure of any potential number is greater than zero; 3. The cardinal number of the standard real number set is equal to the least positive potential infinitelarge p, the unique order p is at least equal to two different cardinal numbers wand c. Since in the order type U2 , continuum hypothesis is an undecidability theorem,that is a simple answer to Hilbert's first problem; 4. For the solution of Newtonian potential with infinitesimal disturbance of 5--function type attached, we get some function terms that can't be found in the standard analysis. Ⅳ The paper solves some historical mathematics problems, for example: 1. Pythagoras, Democritus and Plato put forward successively the concepts of mathematical monad and atom. They made hypothesis that there are indivisible continuums in mathematics. We introduce now potential numbers to solve the hypothesis of Pythagoras, Democritus and Plato; 2. Zeno's general dictum'That,which,being added to another does not make it greater,and being taken away from another does not make it less,is nothing'placed difficult in front of mathematicians for two thousand years or more. The paper solves it in the engative;3. For the hypothesis: 'Aristotle denied that number can produce a continuum, inasmuch as there is no contact in numbers', the paper gives accurate statement and solution; 4. For zhuang Zhou's hypothesisses'No thick may not be integrated 'and 'not be exhausted',the paper gives accurate solutions; 5. The paper proves that the measure of real number set R is equal to zero. Therefore the foundation of the measure theory of sets of real numbers is based on nothing. We must make new base,and proceed a revolution of concept; 6. The paper finds ''the real number set is a discontinuum', because the measure of real number set R is equal to zero then only real number points can not fill up a line. In nowadays mathematical courses of primary school,middle school and university,they claim that the real number axis is just a line. The concept must be changed.