Why do you trust the limit of a sequence of sets more than the limit of cardinal numbers? [on hold]

Question

Let $(s_n)$ be a sequence of sets $s_n$ of rational numbers $q$ such that for $n = 1, 2, 3, ...$
$s_{n+1}$ = $(s_n \cup\ \{q\ |\ n < q \leq n+1\}) \backslash \{q_{n+1}\}$

with $s_1=\{q\ |\ 0 < q \leq 1\}\backslash \{q_1\}$
and $q_1= \frac{1}{1}, q_2=\frac{1}{2}, q_3=\frac{2}{1}, q_4=\frac{3}{1},...$ the positive rational numbers indexed by Cauchy-diagonalization of the matrix of positive rational numbers.

The set $s_n$ contains the rational numbers of the interval $\leq n$ which have not got an index $\leq n$ .
When investigating this case for all natural numbers, we get two limits, one for the sequence of sets and the other for the sequence of cardinal numbers:
$\lim_{n\to \infty}s_n$ = { } is indicating that no rational remains without index.
$\lim_{n\to \infty}|s_n| = \infty$ is indicating that the set of rational numbers without natural index has infinitely many elements, not only for every $s_n$ but also in the limit.

My questions: Why is the first limit considered more reliable than the second one? Has the second limit a mathematical meaning? If so what is it?

Your first «limit» does not make sense to me. What definition of $\lim_{n\to\infty}s_n$ do you use ? Being granted the notion of convergence in use, you only remark that the application $S\mapsto |S|$ is not continuous. Reliability has nothing to do with this, as far as I understand things. — Loïc Teyssier, 9 hours ago
My limit is the usual one. See for instance en.wikipedia.org/wiki/Limsup . Reformulating my question: Why is the sequence of cardinal numbers $(|s_n|)$ accepted to be discontinuous in the limit but not the sequence of sets $(s_n)$ ? — Bacarra, 9 hours ago
@Loic Teyssier The OP just restates the balls in the urn by the Noon paradox (you remember: put in ten, take out one infinitely many times; what is there in the end?) in an awkward way. The only thing that it really shows is that one really needs to define things one is talking about in a clear way to be able to derive logical conclusions, not relying on meaningless "trust". This would be a good example to use in an argument against intuitive calculus and other junk courses if not for the fact that most of people needing convincing would never understand it. — fedja, 8 hours ago
@fedja: I think my posted example is that example which has raised all others. Nevertheless, if there is everything well defined, why is the sequence of sets better trusted than the sequence of "element-numbers"? — Bacarra, 8 hours ago
For the last question: So, the cardinal number of a set is not a continuous functions in the natural topology on the set of subsets in which the first limit is taken. It is surprising, perhaps, but by no means undermining anything. — fedja, 8 hours ago
Why does everything have to be continuous? If you only accept continuous things, how can you accept the existence of the rational numbers AND the real numbers, which in conjunction imply the existence of a function which is nowhere continuous. — Asaf Karagila, 8 hours ago
@Asaf: Here we can apply extended real analysis to the cardinalities. Further we can apply the fact that $s_n$ is never empty for finite $n$ and cannot get empty beyond. — Bacarra, 8 hours ago
No. We can't. $C A R D I N A L S A R E N O T R E A L N U M B E R S .$ $\Large\textbf{CARDINALS ARE NOT REAL NUMBERS.}$ If you can apply real analysis to cardinals, then you can easily apply functional analysis to cooking just because both have a way of "integrating" something. And you can easily apply linear algebra to epidemiology because there you can cross a scalar with a vector to get someone bringing an epidemic from the mountains to the city. — Asaf Karagila, 8 hours ago
The answer to the title question is that you should trust no one over 30. As a corollary, trust no one infinite, hence both sequences and a fortiori their limits are untrustworthy. As another corollary, so am I. — Emil Jeřábek, 8 hours ago
It is accepted for one and only one reason: that is what follows from the definitions. The logic needs to agree with the intuition only on the most basic level. Beyond that it is the intuition that needs to be adjusted to avoid self-contradiction. Math is not law: it needs to be consistent to work. — fedja, 8 hours ago
Please read this answer of mine and the links included there. And also this answer, which perhaps is somewhat clearer on this topic. — Asaf Karagila, 8 hours ago
@Asaf: Of course finite cardinals are real numbers. How could you distinguish the cardinal 2 from the real 2? They were devised by Cantor to be positive integers, even transfinite cardinals. But here we need only the naturas extended by $\infty$ . — Bacarra, 8 hours ago
If you want to think of it in layman terms, think of people on Earth assuming that the population will increase at the current crazy rate, the Universe will be habitable forever, but we'll never get immortal. By the "end of times", the number of people will get larger and larger, but who will be left? This is the topology on the subsets: it cares only about each and every finite subset, but never of the whole. Does it remind you of anything? — fedja, 8 hours ago
@fedja: That does not follow from definitions. Here, by definition, only finite numbers can enumerate rationals. For every finite number we see that it fails to diminish the sequence. Others are not available. That contradict the definition. — Bacarra, 8 hours ago
Finite cardinals are cardinals, which happen to be finite. They are not real numbers, or natural numbers or rational numbers, or integers, or complex numbers. Since Cantor we have progressed quite far to learn new things. If you want, I can insist that the world is made of fire, earth, water and wind. And that all that "atom" mumbo jumbo is nonsense. Have you even seen one? I sure haven't. Certainly these don't exist and all the modern physics is wrong. — Asaf Karagila, 8 hours ago
@Asaf: Cardinals are real numbers. Try to distinguish whether you see here a real or a natural or a cardinal or an ordinal: 1. You cannot. All set theory would become a meaningless game about nothing if the natural numbers could not be defined via cardinals (like Cantor successfully did) or ordinals (by von Neumann). — Bacarra, 8 hours ago
If cardinals are real numbers, what is $\aleph_0-1$ ? why isn't there a set of size $\pi$ ? If cardinals are real numbers, why didn't anyone else notice that in the past 150 years? You make claims, without any backup. Your arguments are nothing more than matheology. — Asaf Karagila, 8 hours ago
@Asaf: The extended real numbers inlcude $\infty$ . There is no problem. During the last 150 years everybody knew that cardinals are real numbers. Your arguments are wrong. — Bacarra, 8 hours ago
Uh, is $\infty$ equals to $\aleph_0$ or $\aleph_1$ or $2^{\aleph_0}$ , or all of them or neither one? Surely you know that set theory proves that $\aleph_0$ equals to neither $\aleph_1$ nor $2^{\aleph_0}$ , and that set theory does not prove (in its usual formulations) that $2^{\aleph_0}=\aleph_1$ , nor it disproves that. I mean, you seem so knowledgeable... — Asaf Karagila, 8 hours ago
@Asaf: Why isn't there a set of size $\pi$ ? Did I say that reals are cardinals? I said cardinals are reals. That is a bit different. Logic! Further we are here in the countable domain only and remain there. That is clear from the example. — Bacarra, 8 hours ago
Yes, but the next step would be to claim that $2^n\to\infty$ and $n\to\infty$ therefore $\aleph_0=2^{\aleph_0}$ , and therefore all infinities are countable. Huzzah. And no, while not all real numbers are cardinals, if you make sense of positive cardinals, either you have to accept the above argument, or you need to say, well $\frac n{2^n}\to 0$ , but then you apply a ratio to cardinals, what does that even mean, is there a set of size $\frac12$ ? And if it does make sense, then by applying limits... we get $\pi$ ! Unless $\pi$ is not the limit of rational numbers, I've forgot that. — Asaf Karagila, 8 hours ago
@Asaf: Please get reasonable. You forget yourself. Finite cardinal numbers are real numbers, have always been so and will remain so. Your claims are nonsense! Therefore I will no longer answer them. — Bacarra, 8 hours ago
You seem to confuse between us, but very well. Good day, Wolfgang. — Asaf Karagila, 8 hours ago
No answer, except a foolish claim. Sad. Or fine? Good luck for your further "research-level research"! — Bacarra, 7 hours ago

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Why do you trust the limit of a sequence of sets more than the limit of cardinal numbers? [on hold]

put on hold as off-topic by Emil Jeřábek, Asaf Karagila, Steven Sam, Stefan Kohl, Yemon Choi 8 hours ago

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Why do you trust the limit of a sequence of sets more than the limit of cardinal numbers? [on hold]

put on hold as off-topic by Emil Jeřábek, Asaf Karagila, Steven Sam, Stefan Kohl, Yemon Choi 8 hours ago

deleted by Todd Trimble♦ 6 hours ago

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