$\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as "what" is to $\beth_1$? So, I recently asked a question about whether $\beth_1$ had a negative, and I was promptly reprimanded because I confused $\aleph_0$ with $\infty$. Therefore, to help me understand the concepts involved better, I have three questions:


*

*It should be obvious (at least, it seems that way to me), that $\infty$ is more related to $\aleph_0$ than it is to any other Cadinal Number, even if it's wrong to say they are the same. That is, $\infty$ would be some form of Countable Infinity, I'd say. The first question is: how exactly are $\infty$ and $\aleph_0$ related? Or am I wrong in my assumption that $\infty$ is of the "Countable Infinity" type?

*Is there anything that is related to $\beth_1$, aka the Cardinality of the Reals, in the same way as $\infty$ is to $\aleph_0$? If so, what would that object be?

*If there is an object satisfying the conditions under (2.), does this object have a negative counterpart, just like $\infty$ has in the form of $-\infty$?


As a final note: please note that I am an interested amateur at Maths, so I am bound to make mistakes. I'm always glad for people pointing out any mistakes I make, but nobody likes being made fun of. First request: please don't make fun of me if I made any mistakes?
Also, I feel it would be a shame if any of these three questions didn't get answered just because one of the other ones contains a mistake. So, second request: as tempting as it might be, try not to focus solely on any mistakes or misconceptions on my part in your answers or comments.
 A: $-\infty, +\infty$ are not cardinal numbers. They are simply some new objects $\notin\Bbb R$ which are adjoined to $\Bbb R$ to yield the extended reals. If you wanted to, you could declare that, OK, $+\infty$ actually is $\aleph_0$, and $-\infty$ is, say, $\{\aleph_0\}$; but you might as well choose $+\infty = \Bbb R$ (which surely is $\notin \Bbb R$) and $-\infty = \{\Bbb R\}$.
Because $+\infty$ comes after $0, 1, 2\dotsc$ in the extended reals, it's natural to think of it as the thing that comes after all of the integers, alias $\omega$, alias $\aleph_0$. But in terms of just order type, notice that any real $r$ has the same claim to fame: $r-1, r-1 + \frac 1 2, r-1 + \frac 2 3, \dotsc, r-1 + \frac n {n+1}, \dotsc$ bears the same relationship to $r$ as $0,1,2\dotsc$ do to $+\infty$.
I can't think of any analog to $\beth_1$ in $\Bbb R$ or related spaces. However, you may find the long line tantalizing and suggestive. It's a construction which uses $\omega_1$ (= $\aleph_1$) and not the more unruly $\beth_1$.
The cardinals don't have "negatives", because infinite cardinal arithmetic doesn't enjoy cancellation: it's not the case that $\kappa + \lambda = \kappa + \mu \rightarrow \lambda = \mu$, because if either $\kappa$ or $\lambda$ is infinite then $\kappa + \lambda = \max(\kappa, \lambda)$. Nevertheless, there is a way to make precise the speculations you're entertaining: the surreal numbers may be just the structure you're looking for.
A: *

*$\infty$ is not a cardinal infinity at all. It is a continuum infinity (my wording, but I'd bet you can find it in some actual sources, too). It was invented for topological reasons, not counting. It's purpose, and that of $-\infty$ is to compactify the real line. We certainly don't introduce them that way - we get enough blank stares as it is - but that is the gist of every reason they became a part of mathematics: To become boundary points at the edges of the Reals. Cardinal infinities are about counting: one-to-one equivalence between sets of objects. There is nothing topological about this. So even though there is a natural identification between the two, they are used in entirely different ways. And when you generalize the concepts to other topological or cardinal infinities, that identification does not generalize.

*Not really. The Surreal numbers provides an "infinity" at this location, but even though the surreals are sort of a continuum, their "infinite" points are ordinals, and thus closely related to cardinals, unlike $\infty$.

*Addition and multiplication on infinite cardinals is trivial and boring, and since the maxim "if $a + c = b + c$, then $a = b$" is often false when infinite values are concerned, subtraction of non-finite values is not well-defined. On the other hand, the cardinals sit in the ordinals, and the ordinals in the surreals, so every cardinal is a surreal number, but the addition and multiplication are different. These operations do have inverses, so there is a $-\aleph_0$ in the surreals (though it is called $-\omega$ instead) and a $-\beth_1$ as well. But this concept has odd variances with what you expect. In particular, there is also an $\aleph_0 - 1$, an $\aleph_0 - 2$, and so on, all of which are separate numbers strictly less than $\aleph_0$, but still greater than any finite number. The surreals are not well-ordered.

