A Note on Iterated Consistency and Infinite Proofs

TL;DR

This paper explores the relationship between iterated consistency, $ ext{Pi}^0_1$-ordinals, and infinite proofs, providing a direct method to determine these ordinals through ordinal analysis.

Contribution

It shows that $ ext{Pi}^0_1$-ordinals can be obtained directly from ordinal analysis using infinite proofs, independent of previous reflection-based approaches.

Findings

$ ext{Pi}^0_1$-ordinals can be read off from infinite proofs.

The approach provides a transparent and direct method for ordinal analysis.

Independence of previous reflection-based methods is demonstrated.

Abstract

Schmerl and Beklemishev's work on iterated reflection achieves two aims: It introduces the important notion of $\Pi^0_1$-ordinal, characterizing the $\Pi^0_1$-theorems of a theory in terms of transfinite iterations of consistency; and it provides an innovative calculus to compute the $\Pi^0_1$-ordinals for a range of theories. The present note demonstrates that these achievements are independent: We read off $\Pi^0_1$-ordinals from a Sch\"utte-style ordinal analysis via infinite proofs, in a direct and transparent way.