o-minimality090607

o-minimality090607 - 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 Abstract. We characterize the o-minimal expansions of the field 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 for o-minimal expansions of any ordered field. 1. PRELIMINARIES. We give a new characterization of o-minimal expansions of the field of real numbers, in particularly transparent mathematical terms. The main definition is that of a rich class ( ). The ( ) indicates that we are working over the field 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 ( ). 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)).
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2 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 field of real numbers is a system ( ,<,0,1,+,•,V), where V obeys MULTIVARIATE ( ) above. (We identify relations with their characteristic functions.) An o-minimal expansion of the field of real numbers is an expansion of the field 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. This is a special case of the more general definition of o-minimal structure introduced in [PS86], which we use in section 4. We prove the following. THEOREM A. Let M = ( ,<,0,1,+,•,V) be an expansion of the field of real numbers. The following are equivalent. i. M is o-minimal. ii. V is a subset of some special rich class ( ). ii. There is a least rich class ( ) containing V, and this rich class ( ) is special. We conclude with a version for arbitrary ordered fields. 2. RICH CLASSES OF REAL FUNCTIONS. The key to proving Theorem A is the characterization of rich classes ( ) in terms of first order definability. We now fix V to be a rich class ( ). Let M = ( ,<,0,1,+,•,V). Let f: n+m . A zero selector of f is a g with the property in the ZERO SELECTION ( ) clause. LEMMA 2.1. There exist f 1 ,...,f 8 : V and f 9 ,...,f 12 : 4 such that i. f 1 (x) = 1/x if x ≠ 0. ii. f 2 (x) = 0 if x ≠ 0; 1 otherwise. iii. f 3 (0) = 0; f 3 (x) = 1 elsewhere. iv. x 0 ´ (f 4 (x) = sqrt(x) f 4 (x) = -sqrt(x)).
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3 v. f 5 (x) = 0 ´ x 0.
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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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o-minimality090607 - 1 WHAT IS O-MINIMALITY by Harvey M...

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