MoL200113: Lee, Troy (2001) Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures. [Report]
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Abstract
Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures Troy Lee Abstract: In this thesis, we show that two definability results of Julia Robinson, namely that multiplication and the successor relation can firstorder define addition, and that the divisibility relation and successor can firstorder define multiplication, also hold over finite structures (where ordering is used instead of successor). The first result is obtained by showing that the BIT predicate can be defined with TIMES and ordering, thus FO(<,BIT)=FO(<,TIMES). Then the ``carrylookahead'' construction can be used to define PLUS with BIT. We also show that there is no firstorder sentence with TIMES but without ordering which can define PLUS. A corollary is then that TIMES cannot firstorder define ordering. Our results, together with recent results on the Crane Beach conjecture show that there is a language definable in FO(<,TIMES) but not in FO(<,PLUS). In other words, there is no firstorder sentence with < and PLUS which can express TIMES.
Item Type:  Report 

Report Nr:  MoL200113 
Series Name:  Master of Logic Thesis (MoL) Series 
Year:  2001 
Date Deposited:  12 Oct 2016 14:38 
Last Modified:  12 Oct 2016 14:38 
URI:  https://eprints.illc.uva.nl/id/eprint/726 
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