The First Order Theory Of Expansions Of O Minimal Structures By The Image Of A Fast Sequence PDF Download

In "Expansions of o-minimal structures by fast sequences", H. Friedman and C. Miller introduced the notion of a fast sequence in o-minimal expansions of the real numbers; they studied the definability theory of expansions of o-minimal structures on the ordered additive group of real numbers by the image of a fast sequence, and they gave several characterizations of the definable sets [J. Symbolic Logic 70 (2005), no. 2, 410]. In such expansions, a fast sequence is an increasing, unbounded sequence such that the growth rate of the associated successor function exceeds the growth rate of any definable unary function restricted to the image of the sequence. This dissertation expands on the work of Friedman and Miller by examining the model theory of the expansion of an o-minimal structure on a linearly ordered group by the image of a fast sequence, including the issue of relative axiomatization. The completeness of the presented axioms follows from a quantifier elimination result for certain interdefinable structures, and the proof of the quantifier elimination result is an adaptation of methods that were used by C. Miller and J. Tyne in "Expansions of o-minimal structures by iteration sequences" [Notre Dame J. Formal Logic 47 (2006), no. 1, 93]. Several consequences of the relative axiomatization are inspired by related results of Friedman and Miller or of Miller and Tyne; for example, the boundary (with respect to the order topology) of every definable unary set of such expansions is a union of finitely many discrete sets. Other consequences include decidability and relative decidability results for specific examples of expansions of an o-minimal structure on a linearly ordered group by the image of a fast sequence.