Cyclic arithmetic (CA) is a first-order logic of numbers, i.e. an arithmetic.
Unlike Peano arithmetic, CA does not have an induction rule. Instead, its proofs are allowed to be coinductively infinite, a.k.a. cyclic.
CA proves exactly the same theorems as PA [Simpson 2017].
Simpson, A. (2017). Cyclic Arithmetic Is Equivalent to Peano Arithmetic. In: Esparza, J., Murawski, A. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2017. Lecture Notes in Computer Science(), vol 10203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-54458-7_17