{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

OnO-minExpansions

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

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

1 ON EXPANSIONS OF O-MINIMAL STRUCTURES PRELIMINARY REPORT by Harvey M. Friedman Department of Mathematics Ohio State University November 23, 1996 [email protected] 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.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}