Alpha-theory: An elementary axiomatics for nonstandard analysis
type: Item
instance of: scholarly article
title: Alpha-theory: An elementary axiomatics for nonstandard analysis
zbMATH Open document ID: 1038.26019
DOI: 10.1016/S0723-0869(03)80038-5
author: Vieri Benci
author: Mauro Di Nasso
published in: Expositiones Mathematicae
publication date: 2004-03-17T00:00:00Z
review text: In the early days of the development of the methods of nonstandard analysis, an interest arose to develop an axiomatic approach. This was, however, to some extent accomplished by the so-called \(\Omega\)-calculus by C.~Schmieden and D.~Laugwitz and later refinements of D.~Laugwitz. This approach is based on adding a new symbol \(\Omega\) (the ghost of an infinitely large number) and postulating that if a ``statement'' about the real number system holds for sufficiently large numbers, then it holds for \(\Omega\). In this paper, the authors present what they call the Alpha-Theory, which presents a beautiful strengthening of the \(\Omega\) theory and has the full strength of Robinson's nonstandard analysis. This is accomplished by a list of five easy to grasp axioms which are given the names Extension Axiom, Composition Axiom, Number Axiom, Pair Axiom, and Internal set Axiom, all formulated in ZFC enlarged with a new symbol \(\alpha\). The \(\alpha\)-calculus is warmly recommended to those who are interested in finding out what Robinson's theory is all about. Even the specialists in the field will find a thing or two to ponder.
Mathematics Subject Classification ID: 26E35
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 2060145
zbMATH Keywords: Omega-calculus
zbMATH Keywords: alpha-calculus
zbMATH Keywords: nonstandard analysis
reviewed by: W. A. J. Luxemburg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2060145