real closed fields are o-miminal
for each odd degree polynomial , has a root
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 .
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.
|Title||real closed fields are o-miminal|
|Date of creation||2013-03-22 13:23:04|
|Last modified on||2013-03-22 13:23:04|
|Last modified by||mathcam (2727)|