$\infty$ and $-\infty$ are to $\aleph_0$ / $\beth_0$ as "what" is to $\beth_1$?

Question

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:

1. 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?
2. 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?
3. 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.

Answer 1

1. $\infty$ 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.
2. 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 $\infty$.
3. 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.

Answer 2

$-\infty, +\infty$ 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.