*(I asked this question on Math.SE earlier but received no response and am therefore moving it here, please note that I realise this question is probably incredibly naïve for the experienced set-theorist, but for an outsider it seems like an important question to ask, and am therefore asking it. )*

Von-Neumann ordinals can be thought of as "canonical" well-orders, indeed every well-order $(W,<)$ has a unique ordinal that is its "order type".

This raises the question of *why* a canonical order is needed, it seems to me that every application of ordinals can be done by using a "large enough" well-ordered set instead that is guaranteed by Hartogs' lemma$^{*}$, for example, instead of performing a transfinite process on an ordinal, we perform it on the "large enough" well ordered set $(X, <)$ whose existence is guaranteed by Hartogs' lemma. Using this method we can prove the first basic applications of ordinals such as Zorn's Lemma$^{\dagger}$ (see for example Asaf Karagila's answer to Zermelo set theory and Zorn's lemma).

$^{*}$ For the purposes of this question let Hartogs' Lemma state: For every set $S$, there exists a well-ordered set $(X, <)$, such that there is no injection from $X\to S$.

$^{\dagger}$ Interestingly popular set-theory books give the exact same argument using ordinals, which are totally superfluous (and need not exist without replacement)!

Remarks/Notes:

The above observations seem to imply, that the "working mathematician" can totally ignore ordinals, but I am more interested in why they are so important to the working

*set*-theorist/logician (given that they literally are a set-theorist's "bread and butter").This is not an entirely useless question that does not "affect things" in any way, since ordinals $\ge \omega+\omega$ need not exist in $\mathsf{ZFC}-\mathsf{Replacement}$, and indeed the above method gives a proof of Zorn's lemma in $\mathsf{ZFC}-\mathsf{Replacement}$. Given that many find replacement dubious, this seems like a strong argument for not using ordinals. (OK, without replacement sets of size larger than $\aleph_{\omega}$ need not exist, but assuming replacement the above method can easily construct such large sets without the need for ordinals.)

I suppose one can ask a similar question about cardinal

*numbers*: Why do we need cardinal numbers, when we can reason about cardinalities using simply injections and bijections on sets?Ordinals seem to give us a "uniform definability" but is that actually useful?

One answer that I have received is "convenience", but if convenience is *the* answer why do we need a formal notion that takes hours to develop when an informal notion seems to suffice (formally)?

gs'Lemma, not Hartog'sLemma, because the guy's name was Friedrich Hartogs. The paper was Über das Problem der Wohlordnung. Mathematische Annalen, 76:590–5, 1915. $\endgroup$there? Why not introduce it earlier, or postpone it until later, or split it into two sections?" There are various topics in set theory that many students balk at, but ordinals are not usually one of them, so I'm sure Jech just put that material where he thought was logical, without worrying about what philosophical implications his choice might seem to convey. $\endgroup$16more comments