Getting Started With Nonstandard Methods - Progress Report
Matt Insall, I. Izyumin · 1985
An important application of logic to mathematics is the development of nonstandard analysis. We study some concepts in this important area and show how to prove various results using nonstandard methods. This progress report is a part of a larger project involving nonstandard meth- ods. The superstructure is a fundamental construct in nonstandard analy- sis. A superstructureV (X) is constructed from some setX by recursively taking power sets. The starting level of a superstructure, denoted by V0(X), is simply the original set X. Any subsequent level, Vn(X), is sim- ply the union ofVn 1(X) and the power set ofVn 1(X). An element is said to have rankn if it is inVn(X) but not inVn 1(X). Superstructures are interesting in that they can be shown to contain any given mathematical entity related to the original set. A formal language is used to formulate sentences that describe such mathematical entities and their properties. The transfer principle dictates that any nonstandard results correspond naturally to standard results, and vice versa. We demonstrate how an informal sentence is translated into a for- mal sentence. We also compute the rank of several familiar entities from linear algebra, such as vector spaces, linear transformations, homomor- phisms, unordered bases, and ordered bases. We then prove a funda- mental theorem about monomorphisms, and proceed to show how vec- tor spaces fit into the nonstandard framework.