Some properties of B\"uchi Arithmetics

Alexander Zapryagaev

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arXiv · 2310.16019

Some properties of B\"uchi Arithmetics

Alexander Zapryagaev

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Abstract

B\"uchi arithmetics $\mathop{\mathbf{BA}}\nolimits_n$, $n\ge 2$, are extensions of Presburger arithmetic with an unary functional symbol $V_n(x)$ denoting the largest power of $n$ that divides $x$. A rank of a linear order is the minimal number of condensations required to reach a finite order. We show that linear orders of arbitrarily large finite rank can be interpreted in $\mathop{\mathbf{BA}}\nolimits_n$. We also prove that the extension of the axioms of Presburger arithmetic with the inductive definition of $V_n$ does not yield an axiomatization of $\mathop{\mathbf{BA}}\nolimits_n$.

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Alexander Zapryagaev. 2023-10-24. Some properties of B\"uchi Arithmetics. https://arxiv.org/abs/2310.16019

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