Reduction of the Number of Quantifiers in Real Analysis through Infinitesimals (Master Thesis, Mathematics Department, California Polytechnic State University, San Luis Obispo)

Details Reduction of the Number of Quantifiers in Real Analysis through Infinitesimals (Master Thesis, Mathematics Department, California Polytechnic State University, San Luis Obispo) Reduction of the Number of Quantifiers in Real Analysis through Infinitesimals (Master Thesis, Mathematics Department, California Polytechnic State University, San Luis Obispo)

2008-09-28

arxiv.org/abs/0809.4814v2

Mathematics

Classical Analysis and ODEs

under the supervision of Todor D. Todorov; 61 pages;

Scientific paper

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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