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UNDEFINABILITY OF MULTIPLICATION IN PRESBURGER ARITHMETIC WITH SETS OF POWERS
Part of:
Model theory
Published online by Cambridge University Press: 10 October 2023
Abstract
We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.
MSC classification
Secondary:
03C15: Denumerable structures
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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