## Non-standard analysis / by Geraldine S. Service

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Service, Geraldine S.

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Logic, Symbolic and mathematicalMathematical analysis

Finiteness principle

Superstructures and monomorphisms

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Show full item record##### 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.