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Extensions of ordered theories by generic predicates

Published online by Cambridge University Press:  12 March 2014

Alfred Dolich ,
Chris Miller  and
Charles Steinhorn
Alfred Dolich*
Affiliation:
Department of Mathematics and Computer Science, Chicago State University, Chicago, IL 60628, USA
Chris Miller
Affiliation:
Department of Mathematics, The ohio State University, Columbus OH 43210, USA, E-mail:miller@math.osu.edu
Charles Steinhorn
Affiliation:
Department of Mathematics, Vassar College, Poughkeepsie NY 12604, USA, E-mail:steinhorn@vassar.edu
*
Department of Mathematics and Computer Science, Kingsborough Community College, 2001 Oriental Boulevard, Brooklyn NY 11235, USA, E-mail:alfredo.dolich@kbcc.cuny.edu
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Extract

Given a theory T extending that of dense linear orders without endpoints (DLO), in a language ℒ ⊇ {<}, we are interested in extensions T′ of T in languages extending ℒ by unary relation symbols that are each interpreted in models of T′ as sets that are both dense and codense in the underlying sets of the models.

There is a canonically “wild” example, namely T = Th(〈ℝ, <, +, ·〉) and T′ = Th(〈ℝ, <, +, · ℚ 〉). Recall that T is o-minimal, and so every open set definable in any model of T has only finitely many definably connected components. But it is well known that 〈ℝ, <, +, · ℚ 〉 defines every real Borel set, in particular, every open subset of any finite cartesian power of ℝ and every subset of any finite cartesian power of ℚ. To put this another way, the definable open sets in models of T are essentially as simple as possible, while T′ has a model where the definable open sets are as complicated as possible, as is the structure induced on the new predicate.

In contrast to the preceding example, if ℝalg is the set of real algebraic numbers and T′ Th(〈ℝ, <, +, ·, 〈alg〉), then no model of T′ defines any open set (of any arity) that is not definable in the underlying model of T.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 2 , June 2013 , pp. 369 - 387
Copyright
Copyright © Association for Symbolic Logic 2013

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References

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