o-minimality110806

# o-minimality110806 - 1 WHAT IS O-MINIMALITY? by Harvey M....

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1 WHAT IS O-MINIMALITY? by Harvey M. Friedman Ohio State University November 8, 2006 We define the crucial notion of o-minimality for expansions of the field of real numbers, in particularly friendly purely mathematical terms. This should help bridge the gap between investigators in o-minimality and mathematicians unfamiliar with model theory, who are concerned with non oscillatory behavior, tame topology, analyzable functions, etcetera. 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 ( ). If f: n m and g: m p V, then h 1 : n p and h 2 : n m+p V, where for all x n , h 1 (x) = g(f(x)). h 2 (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. We will present other choices that can be made for conditions 4,5, which are equivalent in the presence of 1- 3.

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2 The original definition of o-minimality form mathematical logic (model theory) in this context is as follows. We say that f: n m is o-minimal (over the ordered field of reals) if and only if every subset of that is (first order) definable in ( ,<,0,1,+,•,f) is the union of finitely many open intervals and finitely many points. More generally, we say that V is collectively o-minimal ( ) if and only if V obeys MULTIVARIATE ( ) and every subset of that is (first order) definable in the structure M = ( ,<,0,1,+,•,f,g,h,. ..), where {f,g,h,. ..} = V, is a finite union of open intervals and points. We prove the following. THEOREM A. f:R n R m is o-minimal ( ) if and only if f lies in some special rich class ( ). More generally, V is collectively o-minimal ( ) if and only if V is a subset of some special rich class ( ). Furthermore, if V is collectively o-minimal ( ) then there is a least rich class ( ) containing V, and it is special. 1. RICH CLASSES OF REAL FUNCTIONS. Let f: n m . A zero selector of f is a g with the property in the ZERO SELECTION ( ) clause. We now fix a rich class V of real functions. We let M = ( ,<,+,•,f,g,h. ..), where V = {f,g,h,. ..}. LEMMA 1.1. 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)). v. f 5 (x) = 0 ´ x 0. vi. f 6 (x) = 0 ´ x > 0. vii. f 7 (x) = 0 if x 0; 1 otherwise. viii. f 8 (x) = 0 if x > 0; 1 otherwise. ix. f 9 (x,y,z,w) = z if x £ y; w otherwise. x. f 10 (x,y,z,w) = z if x < y; w otherwise. xi. f 11 (x,y,z,w) = z if x ≠ y; w otherwise. xii. f 12 (x,y,z,w) = z if x = y; w otherwise.
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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-minimality110806 - 1 WHAT IS O-MINIMALITY? by Harvey M....

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