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.

  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.
  • $\begingroup$ In relation to your answer to (2.), would there be any circumstances where it would make sense to put a limit (or even an actual value) of a function at something with the "size" of $|| \Bbb R ||$, so to speak? I mean, in the same way limits of functions are often put at $\infty$, which, for lack of a better way of describing what I mean, has the "size" of $|| \Bbb N ||$, aka $\aleph_0$ $\endgroup$ – Ahsim Nreiziev Mar 10 '16 at 4:54
  • $\begingroup$ Only if the function is into the surreals (or other extensions of $\Bbb R$ such as the hyperreals). Otherwise the concept doesn't even make sense, any more than making the limit be "Jefferson Davis" would. However the surreals are a topological space, and they have a number in them that can be identified with $\beth_1$ (in a "natural" manner, though the map isn't all that useful or enlightening). So the concept the limit of a function being $\beth_1$ has a reasonable definition. $\endgroup$ – Paul Sinclair Mar 10 '16 at 5:27
  • $\begingroup$ This is more of a check than an actual question. Checks are usually a lot more stupid than questions -- that's the difference. The Surreal Number that can be identified with $\beth_1$; let's call that $o_{\beth_1}$ for convenience. I assume that if $f(x) = o_{\beth_1}$ and $f(y) = - o_{\beth_1}$, then from this is follows that $f(x) \neq f(y)$, right? By, the way, I told you it was a stupid question -- that's why it's a "check". $\endgroup$ – Ahsim Nreiziev Mar 10 '16 at 17:42
  • $\begingroup$ of course, the only surreal number equal to its opposite is $0$. $\endgroup$ – Paul Sinclair Mar 10 '16 at 17:44
  • 1
    $\begingroup$ I can't make any sense out of that, but surreals, having a huge extension of the reals above $\omega$, also have a huge extension to the reals between $0$ and every real number $> 0$. There is an $1/o_{\beth_1}$ and an $-1/o_{\beth_1}$, but there are smaller surreals yet. When $f(x) = 1/x$, it still holds that $f(+0)$ rises without bound in the surreals. It's just that it has a lot farther to rise. $\endgroup$ – Paul Sinclair Mar 10 '16 at 18:02

$-\infty, +\infty$ 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.

  • $\begingroup$ NOTE My original comment was too long, so I split it into three parts. Sorry if it seems spammy. 1) Yes, I know that $\infty$ and $-\infty$ are not Cardinal Numbers. However, it would seem to me that "$\infty$" would not be uncountable. Therefore, I wondered what the relationship between the non-Cardinal $\infty$ and the Cardinal $\aleph_0$ was. $\endgroup$ – Ahsim Nreiziev Mar 10 '16 at 4:41
  • $\begingroup$ 2) I wasn't looking for an analog to $\beth_1$ in $\Bbb R$, I was looking for a Mathematical "object" which has similar properties in relation to $\beth_1$ as $\infty$ has in relation to $\aleph_0$ $\endgroup$ – Ahsim Nreiziev Mar 10 '16 at 4:41
  • $\begingroup$ 3) I also know that Cardinals don't have negatives. However, $\infty$ does. So, if there would be an $\infty$-like object analog to $\beth_1$. it would make sense to ask if that object has a negative equivalent. $\endgroup$ – Ahsim Nreiziev Mar 10 '16 at 4:42
  • $\begingroup$ @AhsimNreiziev - BrianO's points are about the same as mine. I only answered because I have some different perspectives on them. (1) His point is that while you can consider $\infty$ identified with $\aleph_0$, it is not of any great significance. And the terms "countable" and "uncountable" have nothing to do with $\infty$. (2) that is what the word "analog" means. (3) And he gave a useful answer to you on this. -- In your comments, it seems like you were too busy looking for something to argue about to realize he was just giving you what you were after - as well as he could, anyway. $\endgroup$ – Paul Sinclair Mar 10 '16 at 17:40

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