Authors: Elemer E Rosinger
It has for long been been overlooked that, quite easily, infinitely many {\it ultrapower} field extensions $\mathbb{F}_{\cal U}$ can be constructed for the usual field $\mathbb{R}$ of real numbers, by using only elementary algebra. This allows a simple and direct access to the benefit of both infinitely small and infinitely large scalars, {\it without} the considerable usual technical difficulties involved in setting up and then using the Transfer Principle in Nonstandard Analysis. A natural Differential and Integral Calculus - which extends the usual one on the field $\mathbb{R}$ - is set up in these fields $\mathbb{F}_{\cal U}$ without any use of the Transfer Principle in Nonstandard Analysis, or of any topological type structure. Instead, in the case of the Riemann type integrals introduced, three simple and natural axioms in Set Theory are assumed. The case when these three axioms may be inconsistent with the Zermelo-Fraenkel Set Theory is discussed in section 5.
Comments: 21 Pages.
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[v1] 2014-10-07 05:45:04
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