Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T19:42:17.523Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barwise, Jon and Robinson, Abraham, Completing theories by forcing, Annals of Pure and Applied Logic, vol. 2 (1970), no. 2, pp. 119–142.Google Scholar
[2]Boxall, Gareth and Hieronymi, Philipp, Expansions which introduce no new open sets, this Journal, vol. 77 (2012), pp. 111–121.Google Scholar
[3]Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1990.Google Scholar
[4]Chatzidakis, Zoé and Pillay, Anand, Generic structures and simple theories, Annals of Pure and Applied Logic, vol. 95 (1998), pp. 71–92.CrossRefGoogle Scholar
[5]Dolich, Alfred, A note on definable completeness and weak o-minimality, Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 281–292.CrossRefGoogle Scholar
[6]Dolich, Alfred, Miller, Chris, and Steinhorn, Charles, Structures having o-minimal open core, Transactions of the American Mathematical Society, vol. 362 (2010), pp. 1371–1411.CrossRefGoogle Scholar
[7]Dolich, Alfred, Miller, Chris, and Steinhorn, Charles, Expansions of o-minimal structures by independent sets, in preparation.Google Scholar
[8]van den Dries, Lou, Dense pairs of o-minimal structures, Fundamenta Mathematicae, vol. 157 (1998), pp. 61–78.CrossRefGoogle Scholar
[9]van den Dries, Lou and Lewenberg, Adam H., T-convexity and tame extensions, this Journal, vol. 60 (1995), pp. 74–102.Google Scholar
[10]Fratarcangeli, Sergio, Expansions of o-minimal theories by generic sets, this Journal, vol. 70 (2005), pp. 1150–1160.Google Scholar
[11]Macpherson, Dugald, Marker, David, and Steinhorn, Charles, Weakly o-minimal structures and real closedfields. Transactions of the American Mathematical Society, vol. 352 (2000), pp. 5435–5483.CrossRefGoogle Scholar
[12]Miller, Chris and Speissegger, Patrick, Expansions of the real line by open sets: o-minimality and open cores, Fundamenta Mathematicae, vol. 162 (1999), pp. 193–208.Google Scholar
[13]Pillay, Anand, First order topological structures and theories, this Journal, vol. 52 (1987), pp. 763–778.Google Scholar
[14]Wilkie, A. J., An algebraically conservative, transcendental function, Paris 7 preprints, no. 66, 1998.Google Scholar