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Geometric categories and o-minimal structures
The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts; see, for example, Bierstone and Milman [1]. Regrettably, certain “nice” sets—like { (x, x) : x > 0 } forExpand
Expansions of dense linear orders with the intermediate value property
  • Chris Miller
  • Computer Science, Mathematics
  • Journal of Symbolic Logic
  • 1 December 2001
TLDR
Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R). Expand
Expansions of the Real Field with Power Functions
  • Chris Miller
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 9 June 1994
TLDR
We show in particular that the (O-minimal) theory of the ordered field of real numbers augmented by all restricted analytic functions and all real power functions admits elimination of quantifiers and has a universal axiomatization. Expand
Expansions of o-minimal structures by sparse sets
Given an o-minimal expansion R of the ordered additive group of real numbers and E ⊆ R, we consider the extent to which basic metric and topological properties of subsets of R definable in theExpand
EXPONENTIATION IS HARD TO AVOID
Let 3t be an O-minimal expansion of the field of real numbers. If 31 is not polynomially bounded, then the exponential function is definable (without parameters) in 32 . If 31 is polynomiallyExpand
Structures having o-minimal open core
The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of denseExpand
Expansions of o-minimal structures by dense independent sets
TLDR
We study the structure ( M , ( H ) H ∈ H ) of an o-minimal expansion of a densely ordered group and show that every open set definable in the structure is definably independent in M . Expand
Borel subrings of the reals
A Borel (or even analytic) subring of R either has Hausdorff dimension 0 or is all of R. Extensions of the method of proof yield (among other things) that any analytic subring of C having positiveExpand
A growth dichotomy for o-minimal expansions of ordered fields
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