To an extent, I'm using this MathSE post, primarily including Asaf Karagila's answer, as my point-of-departure. In this case, assimilating cardinal division to division of surreals is not given, so we interpret $\frac{\aleph_0}{\aleph_0} \neq \omega/\omega$. But worse, even though tracking multiplicative back-checking with $\frac{\aleph_{\omega}}{\aleph_n}$ would return $\aleph_{\omega}$, yet complications involving singularity/regularity advise against too quick an identification. So there seems to be something, or a lot of things, about division involving alephs (and other non-aleph transfinite cardinals, in principle, see e.g. Karagila's musings about cofinalities of non-alephs) that testify against the surreal path.

In cardinal arithmetic division is not defined as multiplication gives one result for different products. For instance, $2\aleph_0=\aleph_0$ and $\aleph_0\cdot\aleph_0=\aleph_0$. You cannot divide if different factors still give the same result. These arithmetic rules are not intended to be like numbers. But you can divide surreals just well, they have normal arithmetic.

@Anixx: "You cannot divide if different factors still give the same result"—this comment contradicts the entirety of your answer here, unless there's some distinction I haven't seen yet…

clemens‭ there are different multiplications: cardinal multiplication, which is entirely not like number multiplication, and surreal multiplication, which is a field operation and totally normal, has an inverse operation, division.

@Anixx The whole point of your answer here is that supposedly you can divide if different factors give the same result (e.g. $0·0$ and $0·1$). Now you say you cannot divide if different factors still give the same result. What does this mean?

clemens‭ my bad, I meant that you won't get a good, number-like arithmetic in that case, particularly, there would be no subtraction. But there is no cardinal subtraction anyway. With $0/0=0$ you can still keep subtraction and other usual rules, but it is a special case.

@Anixx: thanks for the clarification!

Anixx‭ but it's not an unexplained statement, "Aleph subtraction/division are ill-defined," it comes from the properties that make such akin to the 0/0 case, hence my interest in adapting wheel theory to this case. Besides, if I'm approaching this from a realist (Aristotelian if not Platonist) direction, squarely dealing with the degeneracy/anomalies of aleph division rather than giving up and adverting to surreal division (which reportedly doesn't help much with analysis anyway) is a more "realist" stance (in the sense of treating the divisibility of a continuum as a question involving the instabilities of zero divisors as well as aleph divisors: I see cardinality, via quantification, as existentially loaded, so if I'm looking for "existent" things, cardinals rather than surreals seem more fitting).

@Ripheus24 Interesting. If I may ask, what do you mean by being "existentially loaded"—and how does it relate to quantification?

clemens‭ the introduction of the existential quantifier from classical/first-order logic was in part established with reference to the analysis of the ontological argument in theology, i.e. Kant's "existence is not a predicate" was understood/reinterpreted as an existence operator not being in predicate position. Then $\exists$ has a sense of at least one, but any cardinal number could be used for a more specific quantifier, so there are ideas about a "countably many" quantifier, or "exactly one," or "exactly two," "at least two," and so on. It's too long and uncertain to explain, but I'm also associating cardinality with spatiality juxtaposed with ordinals :: time, so I'm thinking of the divisibility of space as involving the issues with infinite cardinal division. Like a continuous plenum is "traced" in part in the "negative logical space" of 0/0, $\aleph_0/\aleph_0$, and so on. (Maybe I'm being too influenced by the legend of the Dirac sea...)

Interesting! Wrt. the divisibility of space—and this, I'm thinking, could perhaps somehow be material for another Codidact post—it's occurred to me that what Aristotle was groping towards with his actual/potential infinity distinction has to do with compactness: a closed, bounded interval/region of space is not actually infnite, for Aristotle, because it cannot contain an infinite number of open sets/qualities that simultaneously cover that interval/region. And if you look at the commentators like Avicenna they, curiously enough, say that a really existent line has to be a closed set…

Anyway, not quite on topic, but I was reminded of it by your mention of infinite division of the continuum. With regard to that point—it might not have all the algebraic properties one would desire, but how do you like Sobociński's approach to division by zero?

clemens‭ I have to admit I'm not sure I'm familiar with that one. I have sorted through the ones indicated in the SEP entry on continuity and infinitesimals, and on the Wikipedia page for infinitesimals, and I noticed (IIRC) a remark in the entry on signed zeroes that these could be used as "infinitesimals" in some sense.

I have long analyzed some, usually very simple, formulas involving surreal numbers. I also have read about the surcomplex numbers, so I imagine there are surperplex numbers too. This is daunting, but in turn, I have also long tried to focus on understanding the continuum in terms of the powerset operation on aleph-zero. There are a lot of options, then, like the continuum under L, or in an AD-world, where it is not an aleph, even if it comes from one. My new analysis of this topic is focusing on the question whether it makes much difference to how we think of continuity if we hazard using terms of anomalies of aleph division and cofinality.

clemens‭ then there is a broader "division error gap" when we come to dividing a singular cardinal by one of the regular cardinals that composes it. For example, 0 and 1 are self-cofinal, but cf(2) = 1, and so on, until $cf(\aleph_0) = \aleph_0$, etc. But in the AD-world, $cf(\aleph_3) = \aleph_2$, apparently, so the pattern of fragmentation and defragmentation is different. Either way, the lowest case of singularity in the cofinality sense is over {2, 3, 4, ...} below infinity, but that set is itself infinite as a whole. But they are not continuously defragmented as such, but as of $\aleph_0$, but then if defragmenting the cofinalities of the n > 1 produces a continuous distribution of components, it has to at least go, of course, to $\aleph_1$. Then vs. the AD-world, though, we are tempered against pushing it above $\aleph_2$, at least if we focus on this stipulation about the importance of cofinality.

@Ripheus24 The continuum in AD is not an aleph? Wow! Is it incommensurable with the alephs, then?

Ripheus24‭ in wheel theory subtraction is defined. In cardinals it is not, by definition and cannot be introduced ad all.

Anixx‭ I don't accept statements about possibility and impossibility "by definition." The accepted answer to this question showed a way towards adapting wheel theory to alephs, a way that I see no reason to reject. If you can prove internally to that answer's content that it doesn't work, that will be a helpful response. If you're trying to convince me that aleph division or ratios or whatever are absolutely impossible, I will reject that conviction on the grounds that all sorts of things are definable in mathematics if you try hard enough.

clemens‭ I think that's so, though I've also read something like "a version of CH is derivable using AD," or also where $|R|$ is an immediate successor of $\aleph_0$ in a different direction from $\aleph_1$, so it's a delicate variation. And to be sure, most every time I think I "know" more about set-theoretic transfinite cardinals than I used to, I find out that I know even less in another sense than I did before. This MathOF post about AD is a good look-over of some of the implications, though I shouldn't claim to fully understand what these implications come down to (I can verbally repeat them, and I can "grasp" $cf(\aleph_3) = \aleph_2$ weakly, but beyond that, well...).