  By combining classical results of B\"uchi, some elementary Tauberian theorems
and some basic tools from logic and combinatorics we show that every ordinal
$\alpha$ with $\varepsilon_0\geq \alpha\geq \omega^\omega$ satisfies a natural
monadic second order limit law and that every ordinal $\alpha$ with
$\omega^\omega>\alpha\geq \omega$ satisfies a natural monadic second order
Cesaro limit law. In both cases we identify as usual $\alpha$ with the class of
substructures $\{\beta:\beta<\alpha\}$.
  We work in an additive setting where the norm function $N$ assigns to every
ordinal $\alpha$ the number of occurrrences of the symbol $\omega$ in its
Cantor normal form. This number is the same as the number of edges in the tree
which is canonically associated with $\alpha$.
  For a given $\alpha$ with $\omega\leq \alpha\leq \varepsilon_0$ the
asymptotic probability of a monadic second order formula $\varphi$ from the
language of linear orders is $\lim_{n\to\infty} \frac{\#\{\beta<\alpha:
N\beta=n\wedge \beta\models \Phi\}}{\#\{\beta<\alpha: N\beta=n\}}$ if this
limit exists. If this limit exists only in the Cesaro sense we speak of the
Cesaro asympotic probability of $\varphi$.
  Moreover we prove monadic second order limit laws for the ordinal segments
below below $\Gamma_0$ (where the norm function is extended appropriately) and
we indicate how this paper's results can be extended to larger ordinal segments
and even to certain impredicative ordinal notation systems having notations for
uncountable ordinals. We also briefly indicate how to prove the corresponding
multiplicative results for which the setting is defined relative to the Matula
coding.
  The results of this paper concerning ordinals not exceeding $\varepsilon_0$
have been obtained partly in joint work with Alan R. Woods.

Logic