\begin{abstract}
We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an $\alpha$-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.
\end{abstract}