Finding a limit using arithmetic over cardinals

Question

What is the value of:

$$\lim_{n \to \infty} \frac{n}{2^n} (n \in \mathbb{N})$$

It seems to me that I can use L'Hopital's rule, but does that rule take into account types of infinity? More precisely, it seems to me that the above can be written as

$$ \frac{\aleph_0}{2^{\aleph_0}}$$

By Cantor's theorem, those two infinities are different (in the sense of cardinality). Could I then use some cardinal arithmetic to get the solution, or L'Hopital's rule is just fine? I am probably missing something obvious here though.

Answer 1

No. The limit cannot be written like that. For two reasons. And then there is a third, which is the more important of the three.

1. When considering cardinal arithmetics, the exponent function is not continuous. This means that $\lim_{n\to\infty}2^n\neq2^{\lim_{n\to\infty} n}$. In a clearer statement, $\lim 2^n=\aleph_0$ rather than $2^{\aleph_0}$. So you actually end up with the limit of $\frac{\aleph_0}{\aleph_0}$.

2. Division is not a well-defined concept in cardinal arithmetics. There are no fractions. In real analysis when taking the limit of $\frac{a_n}{b_n}$ we try to test and see how does the ratio plays out as a well-defined number. However for cardinal arithmetics the concept of ratio does not exist. This means that writing $\frac n{2^n}$ is not even a permitted statement in the language of cardinal arithmetics, let alone the limit thereof.

3. And I can't stress how important this part is REAL NUMBERS ARE NOT CARDINALS, AND THE TWO ARITHMETIC SYSTEMS ARE COMPLETELY DISJOINT FROM ONE ANOTHER. I'm sorry if that hurts your eyes, or hard to read. But the point is there. The cardinal arithmetics and the real numbers with their arithmetics have absolutely nothing to do with one another.

   In fact, I am even more careful when I think about these things and the natural numbers are a whole other system from the real numbers and the cardinal numbers and the ordinal numbers. Despite the fact that the natural numbers coupled with the finitary operations can embed very nicely into all three systems (reals, cardinals, and ordinals), it is not the same as any of them.

   It is even more the case when one leaves the realm of finitary operations and moves to infinitary operations which include limits. When the time comes to play with limits, the natural numbers have none; the real numbers have some; the ordinals and the cardinals have them but they behave very differently in either of the systems.

All in all, L'Hospital rule has absolutely nothing to do with cardinals, and the real numbers have nothing to do with cardinals. Thinking that is the first step in the line of many many mistakes that you shouldn't do.

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5. The Aleph numbers and infinity in calculus.
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