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.
Title | real closed fields are o-miminal |
---|---|
Canonical name | RealClosedFieldsAreOmiminal |
Date of creation | 2013-03-22 13:23:04 |
Last modified on | 2013-03-22 13:23:04 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 8 |
Author | mathcam (2727) |
Entry type | Example |
Classification | msc 12D15 |
Classification | msc 14P10 |
Classification | msc 03C64 |
Classification | msc 12D99 |
Related topic | Theory |