Alpha-theory: An elementary axiomatics for nonstandard analysis
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.
scientific article; zbMATH DE number 2060145
scientific article