real closed fields are o-miminal
It is clear that the axioms for a structure to be an ordered field can be written in , the first order language of ordered rings. It is also true that the condition
for each odd degree polynomial
, has a root
can be written in a schema of first order sentences in this language
.
Let be all these sentences together with one that states that all positive elements have a square root.Then one can show that the consequences of are a complete theory .It is clear that this theory is the theory of the real numbers. We call any structure a real closed field (which can be defined purely algebraically also, see here (http://planetmath.org/RealClosed)).
The semi algebraic sets on a real closed field are Boolean combinations
of solution sets of polynomial equalities and inequalities.Tarski showed that has quantifier elimination
, which is equivalent
to the class of semi algebraic sets being closed under projection.
Let be a real closed field. Consider the definable subsets of . By quantifier elimination,each is definable by a quantifier free formula, i.e. a boolean combination of atomic formulas.An atomic formula in one variable has one of the following forms:
- •
for some
- •
for some .
The first defines a finite union of intervals, the second defines a finite union of points. Every definable subset of is a finite union of these kinds of sets, so is a finite union of intervals and points.Thus any real closed field is o-minimal.