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THE NON-AXIOMATIZABILITY OF O-MINIMALITY

Published online by Cambridge University Press:  17 April 2014

ALEX RENNET
ALEX RENNET*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO TORONTO, ONTARIO Email: rennetad@math.utoronto.ca URL: http://www.math.toronto.edu/cms/rennet-alex/
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Abstract

Fix a language extending the language of ordered fields by at least one newpredicate or function symbol. Call an L-structure Rpseudo-o-minimal if it is (elementarily equivalent to) anultraproduct of o-minimal structures. We show that for any recursive list ofL-sentences , there is a real closed field satisfying whichis not pseudo-o-minimal. This shows that the theory To−minconsisting of those -sentences true in all o-minimal-structures, also called the theory of o-minimality (for L), isnot recursively axiomatizable. And, in particular, there are locally o-minimal,definably complete expansions of real closed fields which are notpseudo-o-minimal.

Keywords

O-minimalitylogicmodel theoryaxiomatizabilityultraproducts

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 54 - 59
Copyright
Copyright © Association for Symbolic Logic 2014 

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