There is no smallest infinity in calculus?
Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess.
From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and that infinite numbers are well ordered, $\aleph_1 > \aleph_0$ etc
Now if we move to the world of calculus, even there, there is a difference between one infinity and the other.
$\lim_{x \to \infty} x = \infty$, and $\lim_{x \to \infty} e^x = \infty$, but they are not the same, you could say that the $e^x$ one is bigger, because $\lim_{x\to \infty} \frac{e^x}{x} = \infty$ as can be shown easily with L'Hôpital's rule.
This leads me to believe that unlike in set theory, in calculus there is no smallest infinity, since if $\lim_{x \to \infty} f(x)= \infty$, then $\lim_{x \to \infty} \log (f(x) = \infty$ but a smaller $\infty$.
So which version is "correct"? Is there really a smallest infinity like in set theory? or we can keep getting smaller and smaller to no end like in calculus? Or both are correct in different context? I'm a bit confused.
Which also leads to another question, when we say in calculus that some limit tends to $\infty$, which $\infty$ are we talking about? $\aleph_2$? $\aleph_0$?
There is no such thing as "infinity" in calculus - at least not as an object that is as "concrete" as, say, the number $42$. Things can get as complicated as you want later on, but in the beginning it might be best to think that "$\infty$" is just a symbol and expressions like "$\lim_{x\rightarrow\infty}x^2=\infty$" have a well-defined meaning in the sense that they are just abbreviations for longer sentences in which no more "$\infty$" will occur.
Infinites in set theory are completely different animals, though. Infinities like $\aleph_3$ are special sets which are "singled out" as yardsticks to measure the size (cardinality) of other sets. So, something like $\aleph_3$ really exists according to the axioms of set theory - as opposed to "$\infty$" which - see above - doesn't. (Or to put it more carefully - which has to be given a specific meaning in order to "exist" in a meaningful way. Others have explained ways to do this already.)
There is a reason set theory does not use the symbol $\infty$ - because it would be very easily confused by the way it has been used in limits and calculus in general for a long time. In particular, limits are talking about something that can best be called "linear infinity" or "topological infinity."
Essentially, we can add "values" $-\infty$ and $+\infty$, to the real line, and give them meaning in the sense of convergence, without making them numbers. There is a deep sense in which convergence to these new values is "the same as" convergence to other values on the real line. But the real key is that these two values are not numbers.
The sequences $\{\frac{1}{n}\}$ and $\{e^{-n}\}$ both converge to $0$ at very different rates, but we don't call those zeros different. How fast something converges can be wildly different, but that doesn't mean that the value to which they converge is different.
There is a sense in which convergence can be faster and slower, but that has nothing to do with infinity. You can do the same example above with:
$$\lim_{x\to 0} x = 0 =\lim_{x\to 0} e^{-1/x^2}$$
More generally, $f(x)\to+\infty$ is exactly the same as $\arctan f(x)\to\frac{\pi}{2}$, and similarly, $f(x)\to-\infty$ is exactly the same as $\arctan f(x)\to-\frac{\pi}2$. So the idea that there are different "values" of infinity in calculus is a bit of confusion. In this sense, convergence to "infinity" is the same as convergence to $1$ of a function with range the interval $(-1,1)$.
One typical way to define the real numbers is to define the notion of a Dedekind cut on the rationals, $\mathbb Q$. A Dedekind cut is a subset $C$ of $\mathbb Q$ with the following properties:
- $\forall x\in C\,\forall y\in\mathbb Q(y<x\implies y\in C)$
- $\forall x\in C\,\exists y\in C(x<y)$
- $C\neq \emptyset$
- $C\neq \mathbb Q$.
The set of real numbers is defined as the set of all Dedekind cuts.
But if we defined an Extended Cut with just (1) and (2), then $\emptyset$ corresponds to $-\infty$ and $\mathbb Q$ corresponds to $+\infty$. So in this sense, having the extended real line is "simpler" than having just the real line.
The main reason we start with Dedekind cuts is our obsession with the algebraic properties of $\mathbb R$. We can't define addition on the extended real line in a useful way, for example, because of the problem of $+\infty+(-\infty)$. But if we were only interested in the topological properties of the real line (continuity, limits, etc) then the extended real line actually makes more sense, and it has the advantage that it is simpler.