Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526769 
Title:  Power functions and exponentials in ominimal expansions of fields  
Author:  Foster, T. D. 
ISNI:
0000 0004 2697 1980


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  2010  
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Abstract:  
The principal focus of this thesis is the study of the real numbers regarded as a structure endowed with its usual addition and multiplication and the operations of raising to real powers. For our first main result we prove that any statement in the language of this structure is equivalent to an existential statement, and furthermore that this existential statement can be chosen independently of the concrete interpretations of the real power functions in the statement; i.e. one existential statement will work for any choice of real power functions. This result we call uniform model completeness. For the second main result we introduce the first order theory of raising to an infinite power, which can be seen as the theory of a class of real closed fields, each expanded by a power function with infinite exponent. We note that it follows from the first main theorem that this theory is modelcomplete, furthermore we prove that it is decidable if and only if the theory of the real field with the exponential function is decidable. For the final main theorem we consider the problem of expanding an arbitrary ominimal expansion of a field by a nontrivial exponential function whilst preserving ominimality. We show that this can be done under the assumption that the structure already defines exponentiation on a bounded interval, and a further assumption about the prime model of the structure.


Supervisor:  Wilkie, A. J.  Sponsor:  Engineering and Physical Sciences Research Council  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.526769  DOI:  Not available  
Keywords:  Mathematical logic and foundations ; ominimal ; exponentiation ; power function ; decidability ; model completeness ; real numbers  
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