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Question.

What precise things are known about what Zermelo is hinting at in the below citation? What are scholarly references on Zermelo's own attempts at proving consistency of his axioms? What did Zermelo hope for? Most concretely: are there other publications of Zermelo's on consistency of set theory? Did he lecture on this and if yes, what did he say?

I'm looking for more than "Well, he was hobnobbing with Hilbert in Göttingen and Hilbert's optimism, still unfazed at the time, was giving Zermelo hope for some sort of absolute consistency proofs ...", in particular, looking for a dedicated discussion of how much of the relative-turn (i.e. from a hope of some absolute consistency-proof, in some sense, to the more modest notion of consistency relative to another formal system) was *already recognizable in the documents from the time around 1900, in particular, whether the conjecture kindly provided by Ed Dean in one of the answers below, i.e. whether Zermelo was hoping or planning to work out a relative consistency proof modelled on Hilbert's "Foundations of Geometry", i.e., did Zermelo write about this method of Hilbert's anywhere?

Citation.

In p. 262 of Math. Ann. Vol.65, No. 2 (1908) one can read:

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For convenience, I provide an unidiomatic literal translation:

"[...] of these principles may remain undiscussed here. Even the--certainly very essential--"contradictionlessness" of my axioms I have not yet been able to rigorously prove; rather I have had to restrict myself to occasional remarks that those "antinomies" known today all disappear, if the principles proposed here are adopted. With this [work] I want to at least offer useful preparations to future investigations into such deeper problems."

Remarks.

  • The emphasis on today is mine; evidently Zermelo here is referring to the mundane phenomenon of absence of known problems, the known unknowns, as they say.

  • This question seems appropriate here given the comments of two not entirely unknowledgeable mathematicians to this question.

  • I expect the novelty in all of this to be nil. I do not have illusions that there is anything mathematically new to come of looking into this historical issue. Nor do I want to create a Zermelo-myth along the lines of "Zermelo took a proof of inconsistency of ZFC into the grave" or something like that. Consistency questions are arguably the most studied topic in logic and set-theory, Zermelo's work has been thoroughly digested, and in particular his Math. Ann. 65 paper appears in English translation in volume 1 of his collected works edited by H.-D.Ebbinghaus and A. Kanamori, and Zermelo's mentioning consistency proofs is emphasized, with a (translated) citation here in the first Section.

  • I even expect this very question to have been treated somewhere, but did not search for it. (Isn't this---within reason of course---what Q&A sites are for?)

  • According to the usual narrative, "Hilbert's program" still lay about ten years in the future when this was published.

  • A general recapitulation of the basics on ZFC (in particular the second incompleteness theorem) should perhaps be kept out of this thread. There are many good references on this, on this site and elsewhere. This thread is rather meant to EDIT to clarify this passage: thread is meant to focus on giving a picture of what proof-theory and consistency-proofs meant to mathematicians around 1908, which is more than 20 years before Gödel published the second incompleteness theorem. In particular, are there dedicated historical/mathematical articles on precursors to relative consistency?

  • While I am actively working on something related to (variants of) models of ordinals, whence this question, consistency of ZFC is not an (active) interest of mine; I resolved to ask this nevertheless, because recency can make up for non-novelty, and there seems to be some demand for such a question, and because it sometimes is good to be reminded of, or served with known things, and to complement this question, and to provide a new generation of mathematicians with an occasion to have a (relevant) discussion here. And who knows, maybe something new comes of it?

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    $\begingroup$ Apparently html symbols don't work here :-) $\endgroup$ – Mikhail Katz Jul 13 '17 at 15:18
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    $\begingroup$ @MikhailKatz: Re " I am not sure what the question has to do with Goettingen exactly": it has much to do with Goettingen: Zermelo was at Goetting at the time, Hilbert was, too, and what is conventionally called the "Hilbert school" and the "Hilbert school" in mathematical history all more or less was situated in Goettingen. There is a dose of intentional fancifulness in the title of course. If you think it offensive, I might reconsider and change it, of course. $\endgroup$ – Peter Heinig Jul 13 '17 at 15:25
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    $\begingroup$ True, and Klein was there, too, and Emmy Noether. This does not seem germane to your question. $\endgroup$ – Mikhail Katz Jul 13 '17 at 15:26
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    $\begingroup$ Punning on Sophie Germaine? I will change the title, since the intent is serious. Thanks for the feedback. $\endgroup$ – Peter Heinig Jul 13 '17 at 15:27
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    $\begingroup$ This and your other two Zermelo questions leave it very unclear whether you actually perused the quoted Collected Works, vol. 1. (Over 70 hits each for “(W|w)iderspruch”, “consistency”. This includes, e.g., 1929 Warsaw lectures which sound like an answer to your last question here.) $\endgroup$ – Francois Ziegler Jul 13 '17 at 17:34
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The following remarks may not speak to what you're really after, but given your explicit reference to Hilbert's Program still being years away when wondering what Zermelo might have in mind, they may be somewhat useful.

It's true that at the point of Zermelo's 1908 passage, Hilbert had not yet adopted the more formal approach (along the lines of Frege or Russell-Whitehead) to logical deduction that informed the quest in Hilbert's Program for consistent and complete foundations of mathematics. He didn't have available a notion of syntactic completeness, for instance.

But even so, already in Hilbert's 1899 Foundations of Geometry (pdf here) one finds notions of relative consistency proofs that today's logicians would recognize, despite the informal-by-today's-standards surrounding logical framework, wherein e.g. he essentially shows the independence of an axiom $\varphi$ from a set of axioms $S$ by giving a model for $S+\{\neg\varphi\}$. So it would be well within the realm of possibility that Zermelo in 1908 hoped to devise a relative consistency proof for his axiomatization along similar lines, thus going beyond providing the mere appearance of having addressed the known antinomies that Zermelo references, such as Russell's paradox to which Frege's Grundgesetze had succumbed in 1903.

As to exactly what sort of construction (and relative to what background theory) Zermelo might have envisioned, I can't say (and maybe that's the only part that you're really asking about). But it's also not clear, just from the cited passage at least, that his thoughts even progressed far beyond the point of recognizing the bare desire for such a consistency proof.


While not directly relevant to your particular questions, you might enjoy Reck and Awodey's "Completeness and Categoricity, Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic" (pdf here) for its general discussions of some logical concepts in the era preceding that of Hilbert's Program. Its discussion of Foundations of Geometry has some bearing on my remarks at least.

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    $\begingroup$ In or near 1930, Zermelo published a paper, "Über Grenzzahlen und Mengenbereiche,: in which he exhibited models of set theory, essentially the cumulative hierarchy, over a set of urelements, up to an inaccessible cardinal number of levels. I don't know (1) whether he regarded that as a consistency proof for his axioms and (2) whether he had any ideas like this earlier, for example in 1908. $\endgroup$ – Andreas Blass Jul 13 '17 at 18:10
  • $\begingroup$ @AndreasBlass (1): §5, “We shall not attempt here to provide a logical formal proof of such consistency.” $\endgroup$ – Francois Ziegler Jul 13 '17 at 18:32
  • $\begingroup$ @AndreasBlass: many thanks for this reference. I will try to incorporate this into the summarizing answer I am trying to prepare, but this takes some time, since I am trying to make it informative and keep extrapolating-from-perceived-meaning-of-words to a minimum. For the time being, for readers interested in reading more, this is the article (it is in German) that Andreas cited, and Historia Mathematica, Volume 6, Issue 3, August 1979, Pages 294-304 is very relevant background-reading. $\endgroup$ – Peter Heinig Aug 1 '17 at 6:52
  • $\begingroup$ And, briefly, re "whether he regarded" and re Francois Ziegler's cautioning comment: roughly speaking Zermelo (0) seems not to have considered this to have had any bearing on consistency, in particular in loc. cit. he writes "Es muß vielmehr die Existenz einer unbegrenzten Folge von Grenzzahlen als neues Axiom für die 'Meta-Mengenlehre' postuliert werden, wobei noch die Frage der 'Widerspruchslosigkeit' einer näheren Prüfung bedarf." [My translation: On the contrary, it is necessary to assume the existence of an unbounded sequence of limit numbers, as a new axiom for meta-set-theory, ... $\endgroup$ – Peter Heinig Aug 1 '17 at 7:00
  • $\begingroup$ ... while the the question of consistency needs further investigation.] So, effectively, he seems to be saying that he thinks large-cardinal-axioms necessary, and that he thinks that loc. cit. does not have any bearing on consistency. Rather interestingly though, at the risk of over-dramatizing, Zermelo regarded his 1930 paper to have a bearing on provability, and, to be in contradiction with Gödel's first incompleteness theorem, whence the correspondence in Historia Mathematica, Vol. 6 (3), 1979, 294-304. Gödel painstakingly pointed out to Zermelo where the misunderstandings were. $\endgroup$ – Peter Heinig Aug 1 '17 at 7:07

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