Lukasz
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As I know, such theorem independent on Peano's axioms, is Goodstein Theorem. On the other hand, there in the paper: T. J. Stępień, Ł. T. Stępień, "On the Consistency of the Arithmetic System", J. Math. Syst. Sci. 7, No.2, 43-55 (2017); arXiv:1803.11072 , a proof of consistency of Arithmetic System was published. This proof had been done within this Arithmetic System. Łukasz