example of definable type
Consider as a structure in a language
with one binary relation
, which we interpret as the order.
This is a universal
, -categorical structure (see example of universal structure).
The theory of has quantifier elimination, and so is o-minimal.
Thus a type over the set is determined by the quantifier free formulas over , which in turn are determined by the atomic formulas over .
An atomic formula in one variable over is of the form or or for some .
Thus each 1-type over determines a Dedekind cut over , and conversely a Dedekind cut determines a complete type over .
Let .
Thus there are two classes of type over .
-
1.
Ones where is of the form or for some . It is clear that these are definable from the above discussion.
-
2.
Ones where has no supremum in . These are clearly not definable by o-minimality of .
Title | example of definable type |
---|---|
Canonical name | ExampleOfDefinableType |
Date of creation | 2013-03-22 13:29:43 |
Last modified on | 2013-03-22 13:29:43 |
Owner | aplant (12431) |
Last modified by | aplant (12431) |
Numerical id | 5 |
Author | aplant (12431) |
Entry type | Example |
Classification | msc 03C07 |
Related topic | ExampleOfUniversalStructure |
Related topic | DedekindCuts |