We introduce an abstract framework for treating Goodstein principles [1] via well partially ordered [2] direct limits. Besides termination this approach also yields upper bounds on the proof-theoretic complexity of the underlying processes. This explains, for example, why Goodstein sequences based on the Ackermann function [4] are not more powerful than Goodstein sequences based on hereditary exponential normal forms [3]. (This is part of an ongoing project with Fedor Pakhomov.)

[1] R. L. Goodstein. On the restricted ordinal theorem. The Journal of Symbolic Logic vol. 9 (1944), pp. 33–41.

[2] D. de Jongh, D. H. J.; R. Parikh, Well-partial orderings and hierarchies. Indagationes Mathematicae, Nederlandse Akademie van de Wetenschappen, Proc. Ser. A 80, vol. 39 (1977), no. 3, pp. 195–207.

[3] L. Kirby and J. Paris, Accessible independence results for Peano arithmetic. Bulletin of the London Mathematical Society. vol. 14 (1982), no. 4, pp. 285—293.

[4] A. Weiermann, Ackermannian Goodstein principles for first order Peano arithmetic. Sets and computations, Lecture Notes Series. Institute of Mathematical Sci- ences. National University of Singapore., (S. D. Friedman, D. Raghavan and Y. Yang, editors), vol. 33, World Scientific Publishing Co. Pte. Ltd., 2018, pp. 157–181.