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.