# $\omega$-saturation of $(\mathbb{R},<)$

Could anyone of you explain me why $(\mathbb{R},<)$ is $\omega$-saturated?

EDIT: do you know also why the theory of Boole algebras without atoms is $\omega$-categoric?

Your structure is an endless dense linear ordering, and this is a theory that admits elimination of quantifiers. Thus, every assertion in this language is equivalent to a quantifier-free assertion. If one has finitely many parameters, then the only possible consistent types are the assertions about how those parameters are ordered, and how the new variable $x$ fits into the resulting intervals. That is, the variable $x$ is either equal to one of them, or between two successive ones, or above all or below all of them. All such types are already realized in $\mathbb{R}$, since it has such points already for each of these possible patterns, and so the structure is $\omega$-saturated.
A different approach: the theory of dense linear orderings is $\omega$-categorical.
From that it is easy to see that if a theory is $\kappa$-categorical, then every model of the theory of cardinality at least $\kappa$ is $\kappa$-saturated (just extend the set of parameters to an elementary submodel of cardinality $\kappa$).