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o-minimality113007

# o-minimality113007 - 1 WHAT IS O-MINIMALITY by Harvey M...

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1 WHAT IS O-MINIMALITY? by Harvey M. Friedman* Ohio State University [email protected] http://www.math.ohio-state.edu/%7Efriedman/ November 8, 2006 Revised September 6, 2007 Revised October 29, 2007 Revised November 30, 2007 Abstract. We characterize the o-minimal expansions of the ring of real numbers, in mathematically transparent terms. This should help bridge the gap between investigators in o- minimality and mathematicians unfamiliar with model theory, who are concerned with such notions as non oscillatory behavior, tame topology, and analyzable functions. We adapt the characterization to the case of o-minimal expansions of an arbitrary ordered ring. 1. PRELIMINARIES. We give a new characterization of o-minimal expansions of the ring of real numbers, in particularly transparent mathematical terms. The reader may wish to compare the characterization here with the approach of [Dr98], p.13, which can be adapted to give a characterization in terms of relations instead of functions. The main definition is that of a rich class ( ). The ( ) indicates that we are working over the ring of real numbers. We say that V is a rich class ( ) if and only if 1. MULTIVARIATE ( ). All elements of V are functions f such that the domain of f is some n and the range of f is a subset of some m . 2. POLYNOMIALS ( ). Every polynomial with real coefficients from any n into any m is an element of V. 3. COMPOSITION ( ).

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2 i. If f: n m V, g: m p V, then h: n p V, where for all x n , h(x) = g(f(x)). ii. If f: n m V, g: n p V, then h: n m+p V, where for all x n , h(x) = (f(x),g(x)). 4. ZERO SELECTION ( ). Let f: n+m V. There exists g: n m V such that if f(x,y) = 0 then f(x,g(x)) = 0. We use the adjective “special” if, in addition, we have 5. LIMIT ( ). Every bounded f: V has a limit at infinity. From model theory, an expansion of the ring of real numbers is a system ( ,<,0,1,+,-,•,V), where V obeys MULTIVARIATE ( ). From model theory, an o-minimal expansion of the ring of real numbers is an expansion of the ring of real numbers in which every first order definable subset of is a finite union of open intervals and points. Here and - are allowed as endpoints. Here, and throughout the paper, "definable" means "definable with parameters by a formula in the first order predicate calculus with equality". The above definition is a special case of the more general definition of o-minimal structure introduced in [PS86], which we use in section 4. Specifically, a linearly ordered structure is a system (D,<,V), where (D,<) is a strictly linear ordering, and V is a set of constants from D, multivariate relations on D, and multivariate functions from D into D. An o-minimal structure is a structure M = (D,<,V), where every M definable subset of D is a finite union of open intervals and points, where the endpoints of the intervals lie in D {- , } and the points lie in D. We prove the following.
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