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# Cyclic Proofs for Arithmetical Inductive Definitions

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## Acknowledgements

The authors would like to thank Graham Leigh and Colin Riba for several interesting conversations about (arithmetical) inductive definitions.

## Cite As

Anupam Das and Lukas Melgaard. Cyclic Proofs for Arithmetical Inductive Definitions. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSCD.2023.27

## Abstract

We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain "impredicative" theories; moreover, our cyclic systems naturally subsume Simpson’s Cyclic Arithmetic. Our main result is that cyclic and inductive systems for arithmetical inductive definitions are equally powerful. We conduct a metamathematical argument, formalising the soundness of cyclic proofs within second-order arithmetic by a form of induction on closure ordinals, thence appealing to conservativity results. This approach is inspired by those of Simpson and Das for Cyclic Arithmetic, however we must further address a difficulty: the closure ordinals of our inductive definitions (around Church-Kleene) far exceed the proof theoretic ordinal of the appropriate metatheory (around Bachmann-Howard), so explicit induction on their notations is not possible. For this reason, we rather rely on formalisation of the theory of (recursive) ordinals within second-order arithmetic.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Proof theory
##### Keywords
• cyclic proofs
• inductive definitions
• arithmetic
• fixed points
• proof theory

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## References

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