通过无穷小减少实分析中量词的数量（硕士论文，加州理工州立大学数学系，圣路易斯奥比斯波）
Reduction of the Number of Quantifiers in Real Analysis through Infinitesimals (Master Thesis, Mathematics Department, California Polytechnic State University, San Luis Obispo)
摘要：
我们构造非标准复数（和实数）使用超幂方法的柯西的实数的建设的精神。证明了非标准复数是非阿基米德的代数闭域，非标准实数是全序的实闭非阿基米德域.我们研究了各种类型的非标准数，并发展了$\starr$的非标准完备性结果（饱和原理，有界内集的上确界完备性等）。我们给出了非标准的特征，如开，闭，有界，紧集的单子方面的常见的拓扑对象。我们还考虑了实分析的一些传统问题，如极限、连续性、一致连续性、收敛性、一致收敛性等。在拓扑学和实分析中，我们减少（在某些情况下消除）非标准设置中量词的数量。
英文摘要：
We construct the non-standard complex (and real) numbers using the ultrapower method in the spirit of Cauchy's construction of the real numbers. We show that the non-standard complex numbers are a non-archimedean, algebraically closed field, and that the non-standard real numbers are a totally ordered, real-closed, non-archimedean field. We explore the various types of non-standard numbers, and develop the non-standard completeness results (Saturation Principle, Supremum Completeness of Bounded Internal Sets, etc) for $\starr$. We give non-standard characterizations for such usual topological objects as open, closed, bounded, and compact sets in terms of monads. We also consider such traditional topics of real analysis as limits, continuity, uniform continuity, convergence, uniform convergence, etc. in a non-standard setting. In both topology and real analysis we reduce (and in some cases eliminate) the number of quantifiers in the non-standard setting.
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