Analytic combinatorics of the transfinite: a unifying Tauberian perspective

From a Tauberian perspective we prove and survey several results about the analytic combinatorics of (transfinite) proof-theoretic ordinals. In particular we show how certain theorems of Petrogradskiĭ, Karamata, Kohlbecker, Parameswaran, and Wagner can be used to give a unified treatment of asymptotics for count functions for ordinals. This uniform approach indicates that (Tauberian theorems for) Laplace transforms provide a general tool to establish connections between additive and multiplicative results and may therefore be seen as a contribution to Problem 12.21 in Burris's book on number theoretic density and logical limit laws. In the last section we give applications and prove, in some detail, phase transitions related to Friedman-style combinatorial well-orderedness principles for fragments of first-order Peano arithmetic.From a Tauberian perspective we prove and survey several results about the analytic combinatorics of (transfinite) proof-theoretic ordinals. In particular we show how certain theorems of Petrogradskiĭ, Karamata, Kohlbecker, Parameswaran, and Wagner can be used to give a unified treatment of asymptotics for count functions for ordinals. This uniform approach indicates that (Tauberian theorems for) Laplace transforms provide a general tool to establish connections between additive and multiplicative results and may therefore be seen as a contribution to Problem 12.21 in Burris's book on number theoretic density and logical limit laws. In the last section we give applications and prove, in some detail, phase transitions related to Friedman-style combinatorial well-orderedness principles for fragments of first-order Peano arithmetic.C