ID No.: 2003HZ67003
NAME OF THE STUDENT: Rohit O. Gupta
DISSERTATION TITLE: Nonstandard Models of Peano Arithmetic
ABSTRACT
In my thesis I have tried to explore the relation between logic and mathematics. Generally, contemporary mathematicians are of the view that logic and mathematics are two separate fields and that a mathematician is not interested in the problems of a logician and vice versa. I have tried to show that if we look back into the history of mathematics then we find that much of the development of mathematics and logic goes hand in hand. My main reference for my work has been the book by Richard Kaye Models of Peano Arithmetic’. In the preface of his book Kaye says that Nonstandard models of (first order) Peano arithmetic were first constructed by Skolem (1934).” In the second chapter I give a detailed proof of this theorem which says that nonstandard models of all true first-order L-sentences exist. Kaye in the preface also says that the existence of nonstandard models for arithmetic was implicitly proved by Godel in his incompleteness and completeness theorems. In the third chapter I show how we can prove the existence of nonstandard models for arithmetic using Godel’s theorems. Today the subject of nonstandard models, as Kaye describes, has progressed on the line of Harvey Friedman’s work on initial segments of nonstandard models. In the last chapter I give proof of Friedman’s Theorem, which is considered to be a very important result in this line of research. Kaye describes that there was another line of research into extensions of models of Peans arithmetic and types over such models, starting with the MacDowell-Specker theorem (MacDowell and Specker (1961) and followed by the work of Gaifman (his splitting theorem (1972) and his construction of definable type (1976) and others. In the fourth chapter I have given a brief review on this line of research covering essentially the proof of MacDowell-Specker theorem.
