# example of definable type

Consider $(\mathbf{Q},<)$ as a structure in a language with one binary relation, which we interpret as the order. This is a universal, $\aleph_{0}$-categorical structure (see example of universal structure).

The theory of $(\mathbf{Q},<)$ has quantifier elimination, and so is o-minimal. Thus a type over the set $\mathbf{Q}$ is determined by the quantifier free formulas over $\mathbf{Q}$, which in turn are determined by the atomic formulas over $\mathbf{Q}$. An atomic formula in one variable over $B$ is of the form $x or $x>b$ or $x=b$ for some $b\in B$. Thus each 1-type over $\mathbf{Q}$ determines a Dedekind cut over $\mathbf{Q}$, and conversely a Dedekind cut determines a complete type over $\mathbf{Q}$. Let $D(p):=\{a\in\mathbf{Q}:x>a\in p\}$.

Thus there are two classes of type over $\mathbf{Q}$.

1. 1.

Ones where $D(p)$ is of the form $(-\infty,a)$ or $(-\infty,a]$ for some $a\in\mathbf{Q}$. It is clear that these are definable from the above discussion.

2. 2.

Ones where $D(p)$ has no supremum in $\mathbf{Q}$. These are clearly not definable by o-minimality of $\mathbf{Q}$.

Title example of definable type ExampleOfDefinableType 2013-03-22 13:29:43 2013-03-22 13:29:43 aplant (12431) aplant (12431) 5 aplant (12431) Example msc 03C07 ExampleOfUniversalStructure DedekindCuts