E. Zermelo is widely said to have modelled the (axioms of the) natural numbers by iterating the singleton operation $\{\cdot\}\colon \mathsf{Set}\rightarrow\mathsf{Set}$, $S\mapsto\{S\}$, whence the technical term *Zermelo model*.

J. von Neumann modelled the (axioms of the) natural numbers by iterating the successor operation ${\cdot}^+\colon \mathsf{Set}\rightarrow\mathsf{Set}$, $S\mapsto S\cup\{S\}$,
whence the usual technical term *von Neumann ordinal*.

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Incidentally, they both seem to have started at $S=\emptyset$, with von Neumann only writing `0' in his letter (the use of the Norvegian vowel started about a generation later).
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**Questions.**

(0) Do you know a reference to original work of Zermelo's where the definition via iterates of $\{\cdot\}$ appears?

(1) Are there scholarly references (or possibly, testimonies from people having attended those lectures) about *whether Zermelo defined the natural numbers this way in his lectures* in Göttingen and Zürich?
(It of course seems unlikely that he did, having been much involved in axiomatic set theory, and having received what is now the standard definition from von Neumann.) EDIT: This question now has been copied to a more focused satellite question, which is where it should be answered.

(2) Are there scholarly references about whether Zermelo ever publicly discussed, in lectures or in writing, von Neumann's definition, possibly comparing his definition to his own? EDIT: This question now has been copied to a more focused satellite question, which is where it should be answered.

(3) Are there *mathematico-historical* treatments in specialized journals of these two models? (This is not asking for a discussions, from a modern point of view, of properties of the two models in this thread, see remark below.) EDIT: If one is content with one example, and if a thesis counts as a "specialized journal", this question has been more or less answered by a reference kindly provided by Francois Ziegler in one of the comments.

**Remarks.**

- Motivation for this question comes partly from research, partly from an expository writing project.
- I can read German,
*did*have a look at some original publications of Zermelo's (though not for long...) prior to writing this question, and also did more than just scan for patterns like $\{\{\}\}$, but did not find any trace of the eponymous*Zermelo model*so far. (As I said, I did not read in his papers for very long.) - Von Neumann's definition of the finite ordinals appears at least in a handwritten letter, dated 15. VIII. 1923, that von Neumann sent to Zermelo. This letter is reproduced in facsimile in a German mathematico-historical book by H. Meschowski, partially visible online. So the analogous question appears to be answerable about as nicely as one could possibly imagine. (Of course, it would be somewhat interesting learn that even in this case, Stigler's law of eponomy is validated, but it seems that it is
*not*, and von Neumann ordinals really were first published by von Neumann.) - Zermelo's definition appears to deemed technically inferior to von Neumann's in various well-known aspects (non-transitivity of Zermelo's sets, cardinality of Zermelo's sets being only 0 and 1, non-suitability of Zermelo's sets for defining limit-ordinals, etc). This is an interesting topic in and of itself, possibly fit for another MO-thread, but a bottomless pit and not the topic of this question, which is rather historical and reference-requestish; may the only reflection in this thread of theoretical considerations be through Zermelo himself, see question (3) above.