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.