The famous characterization of countably categorical theories state that such a theory is equivalent with a theory with a finite amount of $n-$types for each $n$, a theory where all models are atomic, a theory which has a saturated elementarily prime model etc. This theorem is cited, in both Chang/Kieslers Model theory, Hodges Model theory and Rothmalers Introduction to model theory, to be a result from 1959 by Engeler, Ryll-Nardzewski and Svenonius all independently. The paper which is cited as Engelers part is the following paper:

Engeler E., A characterization of theories with isomorphic denumerable models, Notices of the American Mathematical Society 6 (1959) 161.

However, when I search on MathSciNet or Zentralblath math for this article nothing appears. Often articles (especially obscure old soviet articles) can not be accessed but are still indexed by these sites, yet this paper (which even belongs to the AMS thus should at least be on MathSciNet) does not appear. When I google the paper, I only find more books and papers which cite it. When I look at Engelers webpage he does not mention it as one of his papers (and he seems to have a quite extensive list).

My question is why? Why can't I find this paper? Why isn't it mentioned anywhere except in references? Where can I find this paper?


A similar question was asked over at MathOverflow. In the comments, Andreas Blass wrote

'Long ago, the Notices of the AMS published essentially just meeting announcements and abstracts of talks. The Calabi citation seems to be well within this "long ago", so it is probably just an announcement of results, not a real paper.'

All of Engeler's papers listed on MathSciNet prior to 1963 are in German. So it seems likely that the Notices "paper" was a brief English-language announcement of a result that appeared in full in a German-language publication (this is supported by the fact that the citation gives just a single page number). Just based on the title, the following paper seems most likely - but maybe someone who has access to the paper (which I do) and speaks German (which I don't) can verify this.

Engeler, Erwin Äquivalenzklassen von n-Tupeln. (German) Z. Math. Logik Grundlagen Math. 5 1959 340–345.

  • 1
    $\begingroup$ I haven't checked the paper itself, but the review on MathSciNet (by Paul Lorenzen) doesn't mention a characterization of countable categoricity. The main theorem described in the review is (what is now called) the omitting types theorem. I'll keep looking in MathSciNet for the right paper. $\endgroup$ – Andreas Blass Nov 7 '17 at 3:19
  • $\begingroup$ Ah, thanks, I should have thought to look at the review! I guess it's possible that Engeler announced the result in the Notices but never published a proof. @AndreasBlass $\endgroup$ – Alex Kruckman Nov 7 '17 at 3:22
  • $\begingroup$ Well, that review was in German, but another Engeler paper, with an English review (by Azriel Lévy), has another ingredient of the result in question. In "Unendliche Formeln in der Modelltheorie" [Z. Math. Logik Grundlagen Math 7 (1961) 154-160] Engeler characterizes countably categorical theories as those in which all models realize the same types. That plus the omitting types theorem gets you a pretty big part of the theorem in the question. $\endgroup$ – Andreas Blass Nov 7 '17 at 3:28
  • $\begingroup$ Ah, good thinking with the paper written in german. When I looked at the 3 Citations of the german paper you mention above I find that 2 of the papers (one is Macpherson's survey of homogeneous structures) mention it as the origin of Engeler's result and the third I don't have acess to, yet mention Ryll-Nardzevski and Svenonius as referenses, thus probably draws the same connection. I still find it kinda strange though that none cites this paper before Macpherson in 2011. Maybe he also got curious when he wrote his paper and looked it up properly. Anyway, this answers my questions, thanks. $\endgroup$ – Ove Ahlman Nov 7 '17 at 12:24

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