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