This item is only available for download by members of the University of Illinois community. Students, faculty, and staff at the U of I may log in with your NetID and password to view the item. If you are trying to access an Illinois-restricted dissertation or thesis, you can request a copy through your library's Inter-Library Loan office or purchase a copy directly from ProQuest.
Permalink
https://hdl.handle.net/2142/71214
Description
Title
On Weak Number Theories
Author(s)
Tung, Shih-Ping
Issue Date
1984
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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.
Use this login method if you
don't
have an
@illinois.edu
email address.
(Oops, I do have one)
IDEALS migrated to a new platform on June 23, 2022. If you created
your account prior to this date, you will have to reset your password
using the forgot-password link below.
