A strong version of Cobham's theorem
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Let k,ℓ≥ 2 be two multiplicatively independent integers. Cobham's famous theorem states that a set X⊆ℕ is both k-recognizable and ℓ-recognizable if and only if it is definable in Presburger arithmetic. Here we show the following strengthening: let X⊆ℕ^m be k-recognizable, let Y⊆ℕ^m be ℓ-recognizable such that both X and Y are not definable in Presburger arithmetic. Then the first-order logical theory of (ℕ,+,X,Y) is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of (ℕ,+,X) is decidable.
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