Is there a notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$?

Question

It is accepted belief $\mbox{countable }\infty$ is not $\mbox{uncountable }\infty$. Is there notion differentiating $\frac1{\mbox{countable }\infty}$ and $\frac1{\mbox{uncountable }\infty}$ (latter should be 'smaller') or in other words does $0$ exist as an entity describable in limits as something given by $\frac1\infty$ in any reasonable sense (in other words are we doing precalculus right way)?

Answer 1

An aside about "belief"

It is accepted belief...

In my opinion, using this phrasing misrepresents what it is mathematicians do in a potentially harmful way. Given a reference/definition, there are things that are defined and things that are not defined. And there are definite claims that are provably true, or provably false, or neither. It shouldn't be a matter of belief.

The grey areas are knowledge (some might not have heard about a definition) and opinions about how far to stretch informal ideas. But clearly defined mathematical ideas do not have this sort of issue. I worry that phrasing things in this way will confuse people and/or encourage people with vague ideas that all they need to do is argue hard enough, rather than make them precise and/or ask (as Brout does well in this question) about ways others have made similar ideas precise.

Limits

in other words...

You appear to be mixing concepts that are defined quite differently.

∞ in Limits

The $\infty$ of limits has essentially two common definitions, and neither of them connect to words like "countable" or "uncountable".

1. The sort of definition most commonly taught in a course like "precalculus" would be that it's just a collection of related shorthands.
   1. ${\displaystyle\lim_{x\to\infty}}f(x)=L$ is a shorthand for something like "for any tolerance $\varepsilon$, there is a point $x=M$ past which $f(x)$ lies within $\varepsilon$ of $L$". For example, ${\displaystyle\lim_{x\to\infty}}\dfrac1x=0$ because to get within, say, $\dfrac{1}{875}$ of $0$, you just have to have $x>875$, etc.
   2. ${\displaystyle\lim_{x\to a}}f(x)=\infty$ is a shorthand for something like "for any cutoff $M$, there is a precision $\delta$ so that if $x\ne a$ is within $\delta$ of $a$, then $f(x)>M$".
   3. Similarly for $-\infty$, one-sided limits at a number $a$, and combining the two, as in ${\displaystyle\lim_{x\to\infty}}f(x)=\infty$.
2. Another style of definition that has the same results for the limits above is given by the extended real number line. The idea is to add on two symbols to the reals, "$\infty$" and "$-\infty$". We usually would then say that $-\infty<a<\infty$ for any real $a$. There are a few equivalent ways of setting this up, but then we should somehow explain how things like limits with $\infty$ and $-\infty$ should work. Long story short, they work the way you would expect with that ordering. In this way, something like ${\displaystyle\lim_{x\to a}}f(x)=\infty$ is no longer a special-case definition; it's like any other limit.

1/∞

The fraction $\dfrac{1}{\infty}$ may be used a couple of different, but closely related ways.

1. It could be a direct shorthand for something like "${\displaystyle\lim_{x\to a}}\dfrac{f(x)}{g(x)}$ where ${\displaystyle\lim_{x\to a}}f(x)=1$ and ${\displaystyle\lim_{x\to a}}g(x)=\infty$". In this case, it's a theorem of real analysis that it will evaluate to $0$.
2. If could refer to a partially-defined division operation on the extended reals. In that case, it's defined to be $0$, but it's defined that way to match up with the answer from the limit interpretation.
3. You wouldn't touch upon this in Precalculus, but if I'm being completely honest, the $\infty$ in an expression like this is also sometimes used for something like an unsigned /undirected version of the other $\infty$ in the shorthands/extended real line. When dealing with the reals, this would be part of the "real projective line". But it's more likely to see this in the context of the complex numbers, where $\infty$ would be a symbol declared by fiat to be near all numbers with large magnitude, inside the "Riemann sphere".

(Un)Countable Infinities

...and Limits

I want to emphasize that none of the three sorts of meanings of $\infty$ related to limits I discussed above have "countable" or "uncountable" attached to them. Even if you made some sort of appeal to justify calling $\infty$ "countable" or "uncountable" for some reason outside what is normally discussed with limits and such, each interpretation I discussed only has one $\infty$. So it would, at best, be either "countable" or "uncountable", and you wouldn't be able to talk about both.

...and Division

When you're talking about uncountable and countable "infinities", what is most often meant are "cardinal numbers" (hereafter "cardinals", since they lack a lot of nice properties numbers usually have). $\infty$ is not usually used to denote any of them. But, being charitable, it's pretty clear what someone might mean by "countable $\infty$": $\aleph_0$, the cardinal that corresponds to countably infinite sets. (There are many different cardinals that correspond to uncountably infinite sets, but I don't think that matters for this discussion.)

Informal rebuttal is what I am going against here

I don't know for sure what will satisfy you, but for cardinals, in pretty much every textbook, as well as Wikipedia and ProofWiki, division is not defined. The Wikipedia link even indirectly explains why it's not defined: the answer to the (cardinal) multiplication problem it's trying to solve may not exist, and when it exists it may not be unique.

Can we try?

Division - Take 2

If division of cardinals is impossible, why do people say things like "these concepts in fact are compatible" and "there is a straightforward way of making sense of it"? And is there any formal definition justifying intuition like "latter should be 'smaller'"?

There are good reasons for saying those things, which makes this a bit subtle.

Now, there is a sense in which you can define whatever you want in math. So I could say "I hereby define the 'Mark' quotient of cardinals to always give the (cardinal) number $17$: $3/4=17$, $\aleph_0/1=17$, etc." But this has no connection to the usual concepts of division or cardinals. So, in my opinion, it's an aesthetically bad definition. Are there any aesthetically better ones?

I can try to make a rough analogy with the negative numbers. Early on, you learned about counting numbers (positive integers) or whole numbers (nonnegative integers), and could add them and sometimes subtract them. But you couldn't evaluate something like $2-5$ because "you can't take away $5$ from $2$". Later on, you were told about/we invented new negative numbers, to make these sorts of calculations defined and consistent.

Unfortunately, to fix division of the cardinals, we need to do much more than simply add in new numbers to be the missing quotients. When dealing with cardinals, we have equations like $2\cdot\aleph_0=\aleph_0$, and a naive approach would end up in a disaster like "dividing both sides be $\aleph_0$ to get $2=1$".

Without getting into too much detail about the cardinals (and the related "ordinals"): To make something that's arithmetically nice enough to allow division, we need to decide not to use the standard operations, and disregard much (but arguably not quite all) of the structure that the cardinals/ordinals usually have. Once this is done, new arithmetic operations can be built, and then any missing "numbers" can be added in. For those who know a lot about ordinals and their (non-commutative!) standard arithmetic, you can read about the necessary new operations on wikipedia.

With the context of the new arithmetic operations and new numbers, the cardinals aren't quite the same thing anymore, because they add and multiply differently. But some things (like how they're ordered) are still true. These copies of the cardinals now live as so-called "surreal numbers".

In the surreals, rather than the cardinals, your intuition/hope is true that $1/\aleph_0$ is larger than $1/\kappa$ where $\kappa$ is (the surreal copy of) an uncountable cardinal. To be clear, neither of those is equal to $0$. And neither of those is intended to be a limit in the sense of limits of real-valued functions like those in precalculus.

Limits - Take 2

Cardinal division is not defined. But mathematicians found a way (which is aesthetically reasonable) to modify the meaning of "cardinal" by rewriting arithmetic and got something interesting for division. Is there some (aesthetically nice) way to connect these fractions back to limits like ${\displaystyle \lim_{x\to \infty}}\dfrac{1}{x}$?

The details are well beyond the scope of this post. Very briefly: If $H$ is any surreal that's "infinite" in the sense that it's larger than every positive integer (including possibilities like the surreal $\aleph_0/2$), then $0$ is the closest real to $1/H$. That sentence has a similar flavor to the meaning of ${\displaystyle \lim_{x\to \infty}}\dfrac{1}{x}=0$. That sort of idea probably can be rigorously generalized to define all of the real number limits if you first work very hard to connect the idea of real number limits to certain systems that allow infinite numbers like this, and then work very hard to squeeze the surreals into that framework.