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A FUNDAMENTAL DICHOTOMY FOR DEFINABLY COMPLETE EXPANSIONS OF ORDERED FIELDS

Published online by Cambridge University Press:  22 December 2015

ANTONGIULIO FORNASIERO
Affiliation:
SECONDA UNIVERSITÀ DI NAPOLI VIALE LINCOLN 5 81100 CASERTAITALYE-mail:[email protected]: http://www.dm.unipi.it/∼fornasiero
PHILIPP HIERONYMI
Affiliation:
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN DEPARTMENT OF MATHEMATICS 1409 W. GREEN STREET URBANA, IL 61801, USAE-mail: [email protected]

Abstract

An expansion of a definably complete field either defines a discrete subring, or the image of every definable discrete set under every definable map is nowhere dense. As an application we show a definable version of Lebesgue’s differentiation theorem.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2015 

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References

REFERENCES

Bruckner, A., Differentiation of real functions, CRM mononograph series, American Mathematical Society, vol. 5, Providence, Rhode Island USA, 1994.CrossRefGoogle Scholar
Dolich, A., Miller, C., and Steinhorn, C., Structures having o-minimal open core. Transactions of the American Mathematical Society, vol. 362 (2010), pp. 13711411.CrossRefGoogle Scholar
Fornasiero, A., Definably complete structures are not pseudo-enumerable. Archive for Mathematical Logic, vol. 50 (2011), pp. 603615.CrossRefGoogle Scholar
Fornasiero, A., Locally o-minimal structures and structures with locally o-minimal open core. Annals of Pure and Applied Logic, vol. 164 (2013), pp. 211229.CrossRefGoogle Scholar
Fornasiero, A. and Servi, T., Definably complete Baire structures. Fundamenta Mathematicae, vol. 209 (2010), pp. 215241.CrossRefGoogle Scholar
Hieronymi, P., Defining the set of integers in expansions of the real field by a closed discrete set. Proceedings of the American Mathematical Society, vol. 138 (2010), pp. 21632168.CrossRefGoogle Scholar
Hieronymi, P., Expansions of subfields of the real field by a discrete set. Fundamenta Mathematicae, vol. 215 (2011), pp. 167175.CrossRefGoogle Scholar
Hieronymi, P., An analogue of the Baire category theorem, this Journal, vol. 78 (2013), no. 1, pp. 207213.Google Scholar
Hieronymi, P. and Tychonievich, M., Interpreting the projective hierarchy in expansions of the ordered set of real numbers. Proceedings of the American Mathematical Society, to appear.Google Scholar
Hrushovski, E. and Peterzil, I., A question of van den Dries and a theorem of Lipshitz and Robinson; not everything is standard, this Journal, vol. 72 (2007), no. 1, pp. 119122.Google Scholar
Lebesgue, H., Sur les fonctions représentables analytiquement.Journal de Mathématiques Pures et Appliquées, vol. 1 (1905), pp. 139216.Google Scholar
Miller, C., Expansions of dense linear orders with the intermediate value property, this Journal, vol. 66 (2001), no. 4, pp. 17831790.Google Scholar
Miller, C., Avoiding the projective hierarchy in expansions of the real field by sequences. Proceedings of the American Mathematical Society, vol. 134 (2006), no. 5, pp. 14831493.CrossRefGoogle Scholar
Miller, C. and Speissegger, P., Expansions of the real line by open sets: o-minimality and open cores. Fundamenta Mathematicae, vol. 162 (1999), pp. 193207.Google Scholar
Oxtoby, J.C., Measure and category. A survey of the analogies between topological and measure spaces, Graduate Texts in Mathematics, vol. 2, Springer-Verlag, New York, 1971.Google Scholar
Rennet, A., The non axiomatizability of o-minimality, this Journal, 2013, to appear.Google Scholar
Riesz, F., Sur l’existence de la dérivée des fonctions monotones et sur quelques problémes qui s’y rattachent. Acta Litterarum ac Scientiarum (Szeged), vol. 5 (1932), pp. 208221.Google Scholar
Rubel, L.A., Differentiability of monotonic functions. Colloquium Mathematicum, vol. 10 (1963), pp. 227279.CrossRefGoogle Scholar
Simpson, S.G., Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, 1998Google Scholar