Non-standard analysis / by Geraldine S. Service

Abstract

This thesis is a study of several theories of Non-standard Analysis. Particular attention is paid to the theories presented by A. Robinson and E. Zakon. Chapter I contains background information from Mathematical Logic and leads to the definition of a Non-standard Model of Analysis.- In Chapter II, we develop the direct product, the ultraproduct and the reduced ultraproduct of a set of similar structures and "construct" a non-standard model of analysis in the form of a reduced ultrapower of the set of real numbers. This model contains genuine "infinite" and "infinitesimal" elements which behave like those which we informally think of in classical analysis. Chapter III contains the theory of Professor Abraham Robinson for first order structures and languages. The Finiteness Principle is applied in the proof of,the existence of Non-standard Models of Analysis. Chapter IV contains the theory of Non-standard Analysis presented by Professor Elias Zakon. This is the main chapter in the paper. His set-theoretical approach is based on the notion of a superstructure which contains all of the set—theoretical "objects" which exist on a set of individuals. A monomorphism is a one-to-one mapping from one superstructure into another superstructure which preserves the validity of sentences. The existence of monomorphisms is proven using ultrapowers. A Non-standard Model of Analysis is defined in terms of a monomorphism. This definition parallels the one given in Chapter I. In Chapter V we define and prove the existence of an Extra-standard Model of Analysis, a concept which is similar to that of a Non-standard Model of Analysis. We also present Professor Robinson's theory for higher order structures and languages. We compare the theories presented by Professors Robinson and Zakon along with ;:hat of Professor M. Shimrat.