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This article states that in 1930 Skolem independently (of Presburger) published quantifier-eliminations for linear algebra over the integers.

I have checked the Ω-Bibliography of Mathematical Logic and was not able to find such a result.

According to Hao Wang's account in Skolem's selected works in logic this is also not the topic of Skolem's "Über einige Satzfunktionen in der Arithmetik" which is quoted in the article. The paper is also from 1931 not 1930.

Does anybody know about this paper of Skolem?

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  • $\begingroup$ My German is a bit rusty, but reviews of Skolem’s paper (Gödel, Ackermann) indicate that it does, indeed, prove quantifier elimination for Presburger arithmetic (albeit in a somewhat unusual language, including function symbols for multiplication by rational constants and $\lfloor x\rfloor$, so apparently the values of terms may be rationals, even though variables range over integers). $\endgroup$
    – Emil Jeřábek
    Commented Mar 30, 2022 at 12:48
  • $\begingroup$ I see @EmilJeřábek I overlooked then that that's equivalent to Presburger feel free to post it as an answer. $\endgroup$
    – user1868607
    Commented Mar 30, 2022 at 12:50

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I don’t have access to Skolem’s treatise, but according to its reviews by Gödel and Ackermann, he proves quantifier elimination for a somewhat unusual system with variables ranging over integers, function symbols for addition, multiplication by rational constants, and $\lfloor x\rfloor$, and the ordering predicate (so, apparently, values of compound terms may be rationals, unlike variables). This is equivalent to Presburger arithmetic.

Concerning the discrepancy in years, the paper is identified as “Skr. Norske Vid.-Akad., Oslo, Math.-Naturv. Kl. 1930, No. 7, 1–28 (1931)”. I interpret this to mean that the paper was published in 1931, but nominally in a 1930 volume.

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