Undecidable extensions of Büchi arithmetic and Cobham-Semënov Theorem

Abstract

AbstractLet k and l be two multiplicatively independent integers, and let L ⊆ ℕn be a l-recognizable set which is not definable in 〈ℕ; +〉. We prove that the elementary theory of 〈ℕ; +, Vk, L〉, where Vk(x) denotes the greatest power of k dividing x, is undecidable. This result leads to a new proof of the Cobham-Semënov theorem.