A Fundamental Dichotomy for definably Complete expansions of Ordered Fields

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.

[1]  Antongiulio Fornasiero.  Locally o-minimal structures and structures with locally o-minimal open core , 2013, Ann. Pure Appl. Log..

[2]  Alex Rennet.  The non-Axiomatizability of O-Minimality , 2014, J. Symb. Log..

[3]  J. C. Oxtoby,et al.  Measure and Category: A Survey of the Analogies between Topological and Measure Spaces , 1971 .

[4]  L. A. Rubel.  Differentiability of monotonic functions , 1963 .

[5]  A. Fornasiero,et al.  Definably complete Baire structures , 2010 .

[6]  Chris Miller,et al.  Structures having o-minimal open core , 2009 .

[7]  A. Bruckner,et al.  Differentiation of real functions , 1978 .

[8]  Philipp Hieronymi.  An analogue of the Baire category theorem , 2013, J. Symb. Log..

[9]  Antongiulio Fornasiero.  Definably complete structures are not pseudo-enumerable , 2011, Arch. Math. Log..

[10]  Ya'acov Peterzil,et al.  A question of van den Dries and a theorem of Lipshitz and Robinson; not everything is standard , 2007, J. Symb. Log..

[11]  Philipp Hieronymi.  Defining the set of integers in expansions of the real field by a closed discrete set , 2009, 0906.4972.

[12]  H. Lebesgue.  Sur les fonctions representables analytiquement , 1905 .

[13]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[14]  Christopher L. Miller.  Expansions of Dense Linear Orders with The Intermediate Value Property , 2001, J. Symb. Log..

[15]  J. C. Oxtoby.  Measure and Category , 1971 .

[16]  Avoiding the projective hierarchy in expansions of the real field by sequences , 2005 .

[17]  Chris Miller,et al.  Expansions of the real line by open sets: o-minimality and open cores , 1999 .