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I've read Skolem's paper on his non-standard models of the arithmetic ("Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen").

I noticed there that he is not mentioning Gödel's incompleteness theorems, though it is known that one of their consequences is the existence of a non-standard model of the arithmetic.

Does anyone know who was the first to derive that connection (between Gödel's theorem and the non-standard model)? I believe that Skolem was aware eventually was Gödel's theorems, no?

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    $\begingroup$ The question is duplicated on Math SE math.stackexchange.com/questions/1280438/… The answer here is only part of Math SE answer, and the part that actually answers the OP questions is missing on hsm. $\endgroup$ – Conifold May 14 '15 at 20:17
  • $\begingroup$ @Conifold, for some strange reason the question at SME was closed. I just voted to reopen. David can do so as well. $\endgroup$ – Mikhail Katz Jun 13 '16 at 7:03
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Some preliminary references :

  • Thoralf Skolem, (1929), Uber einige Grundlagenfragen der Mathematik. Skrifter utgit av Det Norske Videnskaps-Akademi i Oslo I. Matematisk-naturvidenskabelig klasse, 4, pp. 1-49.

  • Kurt Gödel, (1931), Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik 38: 173–98.

  • Thoralf Skolem (1934) Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundam. Math. 23, 150-161.

According to Jens Erik Fenstad :

Towards the end of the 1929 paper Skolem expressed some doubts about the complete axiomatizability of mathematical concepts. His scepticism was based on the set-theoretic relativism which follows from the Löwenheim-Skolem theorem. In 1929 he could give only some partial results, but in a paper from 1934 (and a previous one from 1933) "Über die Nichtcharacterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen" he could prove that there is no finite or countably infinite set of sentences in the language of Peano arithmetic which characterizes the natural numbers. Today, this follows as a simple consequence of Gödel's completeness theorem. The technique used by Skolem was a more direct model-theoretic construction. And this technique, suitably refined to the so-called "ultraproduct" construction, has been an important tool in recent work on model theory.


You can see :

  • Stephen Cole Kleene, Mathematical logic (1967 - Dover ed 2002), page 326-27 for a discussion of Skolem's theorem (1934) on nonstandard models of arithmetic :

It is surprising that the existence of nonstandard models of the usual axioms of elementary number theory was not widely recognized very early by juxtaposing Gödel's completeness theorem 1930 and his incompleteness theorem 1931 [footnote].

[footnote] Not however with quite the version in Gödel 1931, where the incompleteness is given for "Principia Mathematica [Whitehead and Russell, 1910—13] and related systems". PM is not of the form {(first-order) predicate calculus} + {number-theoretic axiom system}. Number-theoretic systems of that form came in with Hilbert 1928, so they are sometimes called Hilbert arithmetic. In 1931-2 (reporting a colloquium held January 22, 1931), Gödel states his incompleteness theorem for such a formal system, with all the usual axioms of elementary number theory.

When the author pointed out the connection between Skolem's and Gödel's theorems in Introduction to metamathematics (1952) p. 430, he knew of no earlier mention of it in the literature. Recently, he has come across 2 lines in Gödel's review of Skolem 1933 [Uber die Unmbglichkeit einer vollstdndigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems] (K.Gödel, 1934, Review of Skolem 1933, Zentralblatt fiir Mathematik und ihre Grenzgebiete, vol. 7, pp. 193-194, p.194 lines 10-11) and 3 in Henkin 1950 (L.Henkin, Completeness in the theory of types, JSL, vol.15, pp. 81-91, p.91 lines 8-10) alluding to such a connection.

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  • $\begingroup$ thanks you. The question is - who first made the connection between Gödel's theorem and Skolem's Models? $\endgroup$ – David May 13 '15 at 13:08
  • $\begingroup$ Mauro, please consider "completing" your answer here (your answer to the exact duplicate over on Mathematics was more elaborate, but after talking to the math mods it was decided that "their copy" should be closed). $\endgroup$ – Danu May 15 '15 at 20:13

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