This paper is intended to give for a general mathematical audience a survey of intriguing connections between analytic combinatorics and logic. We define the ordinals below ε0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible compression technique we give applications to phase transitions for independence results, Hilbert’s basis theorem, local number theory, Ramsey theory, Hydra games, and Goodstein sequences. We discuss briefly universality and renormalization issues (...) in this context. Finally, we indicate how regularity properties of ordinal count functions can be used to prove logical limit laws. (shrink)

The termination of rewrite systems for parameter recursion, simple nested recursion and unnested multiple recursion is shown by using monotone interpretations both on the ordinals below the first primitive recursively closed ordinal and on the natural numbers. We show that the resulting derivation lengths are primitive recursive. As a corollary we obtain transparent and illuminating proofs of the facts that the schemata of parameter recursion, simple nested recursion and unnested multiple recursion lead from primitive recursive functions to primitive recursive functions.