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I have now claimed a few times on the internet, based on something (sensible!) I read, that at some point in the 1920s, that Zermelo at one point considered as a set theoretic axiom (schema) something equivalent to a restricted version of Replacement, corresponding to a function with countable domain, before going all out by 1930. Now, however, I cannot find the document in which I read this, and I cannot remember anything else about it! I foolishly didn't write this down. I have consulted Ebbinghaus' book, and I can't see it in there. I have consulted Kanamori's edition of Zermelo's collected works, and I can't see it in the prefatory essays on Zermelo's 1929 or 1930 set theory papers.

However, and this is where my faulty memory shows up even more: I also in one place claimed it was Fraenkel who considered countable Replacement! I cannot find this either. And in trying to hunt down my original source, I recognised the phrase

it is also to be noted that Skolem only wrote that "we could introduce" Replacement

in Kanamori's 2012 In Praise of Replacement. But I didn't see, on a search through there, anything corresponding to my half-remembered claim.

Can anyone enlighten me?

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  • $\begingroup$ Given that your preliminary research has yielded no leads, I would expect this to turn into a challenging quest. Is there any contextual information that might help us? For example, did you come across this information in a blog, a journal, a book? Did you hear about it from a colleague (at a particular conference perhaps, or at a particular institution)? Were you researching a particular topic when you encountered this information? Based on personal experience: False memories are a thing; they can be re-enforced by repeated telling. How confident are you (percentage) about the Zermelo origin? $\endgroup$
    – njuffa
    Dec 4, 2022 at 12:54
  • $\begingroup$ It was in a book, I'm pretty sure. I was looking into the history of Replacement and/or the cumulative hierarchy, as I believe I was looking into Borel Determinacy at the time. It's not so much as it's a false memory: I wrote comments online at the time pointing out this interesting fact I'd just read. So either I or the author made a slip, or this is out there somewhere. $\endgroup$
    – David Roberts
    Dec 4, 2022 at 20:43
  • $\begingroup$ That is helpful context. Could the following be related? G. Kreisel, "Logical Aspects of the Axiomatic Method: On Their Significance in (Traditional) Foundations and in Some (Now) Common or Garden Varieties of Mathematics", In: H.-D. Ebbinghaus, et al. (eds.), Logic Colloquium '87, (Granada, Spain, July 1987), Elsevier 1989, pp. 183-217. On p. 213: " The second example concerns (the power set operation, and) the replacement axiom, which Zermelo omitted in 1908, or, more specifically, transfinite recursion, which Fraenkel and von Neumann considered in the twenties, and Borel determinacy. " $\endgroup$
    – njuffa
    Dec 4, 2022 at 21:50
  • $\begingroup$ Also, at the point in time that I was reading this (July last year), I wrote: "Fraenkel, after the work where he (apparently tentatively!) suggested Replacement as a strengthening of Zermelo’s axioms, also proposed an alternative axiom equivalent to countable Replacement." So perhaps this is his 1925 paper? My mathematical German is far too weak to skim these papers and look for key words. $\endgroup$
    – David Roberts
    Dec 4, 2022 at 22:11
  • $\begingroup$ Matthew Foreman and Akihiro Kanamori (eds.), "Handbook of Set Theory, Vol. 2", Springer 2010. Page 12: " In the 1920s, fresh initiatives in axiomatics structured the loose Zermelian framework with new features and corresponding axioms, the most consequential moves made by John von Neumann [...] in his doctoral work [...] Fraenkel and [...] Skolem had independently proposed Replacement to ensure that a specific collection resulting from a simple recursion be a set, but it was von Neumann's formal incorporation of transfinite recursion as method which brought Replacement into set theory. " $\endgroup$
    – njuffa
    Dec 4, 2022 at 22:26

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