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Cyclic arithmetic is equivalent to Peano arithmetic. (English) Zbl 06720996
Esparza, Javier (ed.) et al., Foundations of software science and computation structures. 20th international conference, FOSSACS 2017, held as part of the European joint conferences on theory and practice of software, ETAPS 2017, Uppsala, Sweden, April 22–29, 2017. Proceedings. Berlin: Springer (ISBN 978-3-662-54457-0/pbk; 978-3-662-54458-7/ebook). Lecture Notes in Computer Science 10203, 283-300 (2017).
Summary: Cyclic proof provides a style of proof for logics with inductive (and coinductive) definitions, in which proofs are cyclic graphs representing a form of argument by infinite descent. It is easily shown that cyclic proof subsumes proof by (co)induction. So cyclic proof systems are at least as powerful as the corresponding proof systems with explicit (co)induction rules. Whether or not the converse inclusion holds is a non-trivial question. In this paper, we resolve this question in one interesting case. We show that a cyclic formulation of first-order arithmetic is equivalent in power to Peano arithmetic. The proof involves formalising the meta-theory of cyclic proof in a subsystem of second-order arithmetic.
For the entire collection see [Zbl 1360.68010].

68Qxx Theory of computing
Full Text: DOI
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