Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

Andreas Weiermann
2005 Annals of Pure and Applied Logic  
This paper is intended to give for a general mathematical audience (including non logicians) a survey about intriguing connections between analytic combinatorics and logic. We define the ordinals below ε 0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give applications to phase transitions for independence results, Hilbert's basis theorem, local number
more » ... ry, Ramsey theory, Hydra games, 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.
doi:10.1016/j.apal.2005.05.012 fatcat:dxkq6aj46ffmhkupj6mmhrqazq