A ring homomorphism is enough to get nonstandard analysis
The paper contains an exposition of some fundamental facts of nonstandard analysis based on a special class of ring homomorphisms. The principal notion of the paper is a {good homomorphism}; this is a special surjective homomorphism of \(\mathbb{R}^I\) where \(I\) is a set on a field \(\mathbb{F}.\) By using incomplete good ultrafilters [\textit{H. J. Keisler}, Ann. Math. (2) 79, 338--359 (1964; Zbl 0137.00803), \textit{K. Kunen}, Trans. Am. Math. Soc. 172(1972), 299--306 (1973; Zbl 0263.02033)], it is proved that for every infinite cardinal there exists a good hyper-homomorphism \(\phi:\mathbb{R}^I\rightarrow \mathbb{F}\). It is proved that under the hypothesis \(2^k=k^+\), where \(k\) denotes the succesor cardinal for \(k,\) all hyperreal fields originating from good hyper-homomorphisms on rings \(\mathbb{R}^I\) are isomorphic as ordered fields.
scientific article; zbMATH DE number 2118937