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  1. T-Height in Weakly O-Minimal Structures.James Tyne - 2006 - Journal of Symbolic Logic 71 (3):747 - 762.
    Given a weakly o-minimal theory T, the T-height of an element of a model of T is defined as a means of classifying the order of magnitude of the element. If T satisfies some easily met technical conditions, then this classification is coarse enough for a Wilkie-type inequality: given a set of elements of a model of T, each of which has a different T-height, the cardinality of this set is at most 1 plus the minimum cardinality of a set (...)
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  • Expansions of o-Minimal Structures by Fast Sequences.Harvey Friedman & Chris Miller - 2005 - Journal of Symbolic Logic 70 (2):410-418.
    Let ℜ be an o-minimal expansion of (ℝ, <+) and (φk)k∈ℕ be a sequence of positive real numbers such that limk→+∞f(φk)/φk+1=0 for every f:ℝ→ ℝ definable in ℜ. (Such sequences always exist under some reasonable extra assumptions on ℜ, in particular, if ℜ is exponentially bounded or if the language is countable.) Then (ℜ, (S)) is d-minimal, where S ranges over all subsets of cartesian powers of the range of φ.
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  • Expansions of the Real Field by Open Sets: Definability Versus Interpretability.Harvey Friedman, Krzysztof Kurdyka, Chris Miller & Patrick Speissegger - 2010 - Journal of Symbolic Logic 75 (4):1311-1325.
    An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ (...)
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  • Definably Connected Nonconnected Sets.Antongiulio Fornasiero - 2012 - Mathematical Logic Quarterly 58 (1):125-126.
    We give an example of a structure equation image on the real line, and a manifold M definable in equation image, such that M is definably connected but is not connected.
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