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We construct the non-standard complex (and real) numbers using the ultrapowermethod in the spirit of Cauchy's construction of the real numbers. We show thatthe non-standard complex numbers are a non-archimedean, algebraically closedfield, and that the non-standard real numbers are a totally ordered,real-closed, non-archimedean field. We explore the various types ofnon-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 topologicalobjects as open, closed, bounded, and compact sets in terms of monads. We alsoconsider such traditional topics of real analysis as limits, continuity,uniform continuity, convergence, uniform convergence, etc. in a non-standardsetting. In both topology and real analysis we reduce (and in some caseseliminate) the number of quantifiers in the non-standard setting.

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