Timeline for answer to Why Zermelo postulated the existence of a set with no finite limit to the ranks of its elements? by Ali Enayat
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| Dec 7, 2025 at 19:00 | comment | added | Zuhair Al-Johar | It is nice to know that this was the origins for Zermelo's infinite set. But, really the informal idea of Dedekind seems to be loose, for I can say that the thought s' saying that $S$ is an object of our thought, leads to s' being an element of S, since clearly $s'$ is not the same as $S$, which is saying that $\{S\} \in S$ and so $S$ is a set, which is paradoxical. | |
| Dec 7, 2025 at 17:54 | comment | added | Robin Saunders | @ZuhairAl-Johar Perhaps it's weaker in one sense, but it's also stronger in another. After all, the axioms being compatible with the existence of urelements is very different from implying that there are any. (There is always a set whose elements are urelements: the empty set.) Without actively assuming the existence of urelements, it's consistent that there are none and hence every set with finite rank is finite. So in this sense, it's not weaker, just different. | |
| Dec 7, 2025 at 17:06 | comment | added | Zuhair Al-Johar | @RobinSaunders, the point is that sets of urelements could have been used to construct a finitely ranked infinite set, which in a sense seems to be weaker than jumping to an infinite rank. Infinity is basically about width (i.e. cardinality), not about height (rank). | |
| Dec 7, 2025 at 2:59 | comment | added | Robin Saunders | @ZuhairAl-Johar perhaps rather than "allows for", it would be better to say "is compatible with". The formal mathematical content of the statement remains the same, while the language makes it sound less like a matter of design which demands attention. | |
| Dec 6, 2025 at 20:30 | comment | added | Zuhair Al-Johar | @StevenStadnicki, I see your point. But, if it allows them, then one must make use of them, otherwise this allowance would be superfluous. One potential use of them is that one can construct infinite sets with finite ranks. By the way, Zermelo's set theory proves existence of a set of urelements (if the term subsets includes urelements). | |
| Dec 6, 2025 at 18:06 | history | edited | Ali Enayat | CC BY-SA 4.0 |
fixed grammar
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| Dec 6, 2025 at 17:55 | comment | added | Steven Stadnicki | @ZuhairAl-Johar Nothing in Zermelo's axioms says anything about either ur-elements or their collection. You're correct that it 'allows for' ur-elements, but it doesn't mandate them. Given that it's explicitly a theory of sets, not of a (potential) set-theoretic universe, it makes sense to say as little about any potential ur-elements as possible. | |
| Dec 6, 2025 at 17:24 | history | edited | Ali Enayat | CC BY-SA 4.0 |
Elaborated the answer.
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| Dec 6, 2025 at 14:59 | vote | accept | Zuhair Al-Johar | ||
| Dec 6, 2025 at 14:57 | comment | added | Zuhair Al-Johar | I see! So that was the reason why. But, looking backwards, it appears to me that matters could have be done by taking the set A of all urelements to be a Tarski infinite set in the sense of not being well order-able by a relation whose converse is a well ordering on it also, then the set of all equivalence classes of finite subsets of A under equivalence relation bijection, could have served as a demonstration of Dedekind's infinity. There is no need to invoke an infinitely ranked set for that purpose. | |
| Dec 6, 2025 at 13:31 | comment | added | Ali Enayat | @ZuhairAl-Johar A specific set has the advantage of giving motivation for a general existential assertion of the type you advocate. Dedekind used this specifc set to motivate his definition of "Dedekind infinite" the same essay; and Zermelo was fully aware of this, given Dedekind's well-established stature and influence, especially on German-speaking mathematicians. | |
| Dec 6, 2025 at 13:12 | comment | added | Zuhair Al-Johar | Yes, but that's different from simply having a set with infinitely many elements? One could have straightforwardly say that there is a Dedekind infinite set without posing a specific set like this. The question is about the infinite rank. | |
| Dec 6, 2025 at 12:30 | history | answered | Ali Enayat | CC BY-SA 4.0 |