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# Mathematics - Theses, Dissertations, and other Required Graduate Degree Essays

Receive updates for this collection## An application of non-standard model theoretic methods to topological groups and infinite Galois theory.

The purpose of this paper is to review some of the work done by Abraham Robinson in topological groups and infinite Galois Theory using ultrapowers as our method of obtaining non-standard models. Chapter One contains the basic logical foundations needed for the study of Non-Standard Analysis by the method of constructing ultrapowers. In Chapter Two, we look at non-standard models of topological groups and give the characterizations of some standard properties in non-standard terms. We also investigate a non-standard property that has no direct standard counterpart. In Chapter Three, we analyze an infinite field extension of a given field r and arrive at the correspondence between the subfields of our infinite field that are extensions of r and the subgroups of the corresponding Galois group through the Krull topology by non-standard methods.

## Three contributions to the theory of recursively enumerable classes.

Priority arguments are applied to three problems in the theory of rce. classes. Chapter I: A conjecture of P. R. Young in A Theorem on Recursively Enumerable Classes and Splinters, PAMS 17,5 (1966), pp. 1050-1056, that an r.e. class can be constructed with any pre-assigned finite number of infinite r.e. subclasses, is answered in the affirmative. Chapter II: Standard classes and indexable classes were introduced by A. H. Lachlan (cf. On the Indexing of Classes of Recursively Enumerable Sets, JSL 31 (1966-),, pp. 10-22). A class C- of r.e. sets is called sequence enumerable if the r.e. (3- is indexable => fl. is subclass enumerable, but none of the implications can be reversed. Chapter. Ill: A partially ordered set (&,<} is represented by the r.e. class G- if (&,<} is isomorphic to (C-,E). Sufficiently many p.o. sets are proved representable to verify a conjecture of A. H. Lachlan that representable p.o. sets and arbitrary p.o. sets are indistinguishable by elementary sentences.