We explore the idea of using automatic and similar kind of presentations of structures to deal with the conceptual problem of natural proof-theoretic ordinal notations. We conclude that this approach still does not meet the goals.

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Beklemishev et al. (Sun,) studied this question.

synapsesocial.com/papers/68e60662b6db643587599d26 https://doi.org/https://doi.org/10.48550/arxiv.2407.10198

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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

1Proofs that Modify Proofs2024
2Ordinal Well-Foundedness as a Structural Termination Criterion for Recursive Systems2026
3Well quasi-order and atomicity for combinatorial structures under consecutive orders2025
4Well quasi-order and atomicity for combinatorial structures2026
5The Algebra in Natural Structures2026