OnO-minExpansions

OnO-minExpansions - 1 ON EXPANSIONS OF O-MINIMAL STRUCTURES...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 5

OnO-minExpansions - 1 ON EXPANSIONS OF O-MINIMAL STRUCTURES...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online