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Title:  On Weak Number Theories 
Author(s):  Tung, ShihPing 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Decidability and definability are two separate but quite related topics in logic. Many undecidability results are proved by positive definability results. In Chapter 1 we reformulate Schinzel's theorem about diophantine equations with parameters to get some number theoretic results. In later chapter we apply these results to solve various decidability and definability problems. In Chapter 2 we prove that (FOR ALL)('n)(THERE EXISTS) over Z is decidable, then we generalize this result to an arbitrary ring of integers of a finite extension of rational numbers. In Chapter 3 we give a necessary condition for a set to be (FOR ALL)('n)(THERE EXISTS)diophantine definable over R. From this necessary condition we can show that many subsets of R including N and cofinite subsets, are not (FOR ALL)('n)(THERE EXISTS)diophantine definable. We also characterize those subsets of N such that the set and its complement in N both are (THERE EXISTS)diophantine definable over N. From this we can answer negatively the question asked by J. P. Jones {5}. In Chapter 4 we prove that the set of prime numbers cannot be defined by a formula containing but one quantifier ranging over N. So far this is the only definite subset of N we know which has this property. Z. Adamowicz constructed a model which has some induction schemes but Matijasevic's theorem fails in this model. In order to prove her induction schemes she has to assume a very strong unproved conjecture, namely Schinzel's hypothesis H. With a result we prove in Chapter 1 we can prove the same induction schemes without assuming hypothesis H. 
Issue Date:  1984 
Type:  Text 
Description:  58 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1984. 
URI:  http://hdl.handle.net/2142/71214 
Other Identifier(s):  (UMI)AAI8409843 
Date Available in IDEALS:  20141216 
Date Deposited:  1984 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois