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1 ON EXPANSIONS OF O-MINIMAL STRUCTURES PRELIMINARY REPORT by Harvey M. Friedman Department of Mathematics Ohio State University November 23, 1996 friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ An o-minimal structure is any relational structure in any relational type in the first order predicate calculus with equality, where one symbol is reserved to be a dense linear ordering without endpoints, satisfying the following condition: that every first order definable subset of the domain is a finite union of intervals whose endpoints are in the domain or are ± . First order definability always allows any parameters, unless explicitly indicated otherwise. Fix M = (R,<,…) to be an o-minimal structure. We say that E has property * over M if and only if E R and the following holds: Let f1,…,fr:I Æ R be definable over M, where I is an interval with endpoints in R, where each fi is strictly monotone, and where for all x I, f1(x),…,fr(x) all disagree. Let a1,…,ar {0,1}. Then there exists x I such that for all 1 £ i £ r, fi(x) E if and only if ai = 1. Let M(E) be the result of expanding M by a unary predicate symbol for membership in E, where E has property * over M. We want to study M(E). We will now show that M(E) has elimination of quantifiers in the following sense. We assume that M has symbols for every M definable function from every Cartesian power Rk into R, including k = 0 (i.e., constants). It is convenient to let 0 be an arbitrary element of R. Thus we will consider only atomic formulas of the form F(x1,…,xk) = 0 and F(x1,…,xk) E. There is no need to consider any other atomic formulas. We want to prove that every formula is equivalent to a Boolean combination of atomic formulas with no new free variables.
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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