What ordinal number is "the usual real" $\infty$ We talk about infinity (denoted with symbol $\infty$) when talking about the real numbers. For example the intervals $(-\infty, a]$, $[b, \infty)$. I guess $\infty$ can be understood to be the point added to $\mathbb{R}$ when doing the one point compactification, but then $-\infty = \infty$ (?).
Is it some ordinal number and if yes, what?
To my understanding, the point $\infty$ is just a symbol and can be like what ever, just a point that gets added. But is there a way to make a connection with the ordinal arithmetic? It could be $\omega$ (the countable infinity) since that is larger than any natural number and hence larger than any real number, but how about all the rest real numbers, what ordinal numbers are they? Then again, I think it should be the ordinal of continuum ($\cal{P}(\mathbb{N})$??). I sense this has something to do with well-ordering.
How about when we deal with limits of type $x\to \infty$, is there some ordinal stuff that applies there, or are these two realms ($\mathbb{R}$ and the ordinal numbers) completely separate?
 A: It's not an ordinal. It is a formal notation for "larger that any real number". But it is a "vague concept" in how we use it (e.g. $\infty^2$ or $\infty/\infty$, for example). We sometimes use it as a limit of a sequence, where the index set is $\Bbb N$, and sometimes a limit of a function over the real line, where the index set is $\Bbb R$. And arguably, we can always move from one to another, but that's not the point.
Ordinals are defined as isomorphism types of well-ordered sets, or in the von Neumann ordinal assignment: canonical representatives for each isomorphism class. 
So while we can sometimes think about $\lim_{n\to\infty}x_n$ as a sequence indexed by $\omega$, and therefore argue that $\infty$ is somehow $\omega$, this is not entirely accurate and does not capture the idea of $\infty$ in the real line properly.
To separate the terms, $\infty$ is a potential infinity. It is a notion of infinity which simply tells you that you can grow arbitrarily large. $\omega$, and transfinite ordinals (and cardinals for that matter), is an actual infinity. This means that this is a notion which corresponds to a specific set which exists (in the standard mathematical universe, anyway).
A: Infinity is just a concept and $\infty$ represents this concept. $\omega$ is an ordinal defined as being the smallest infinite ordinal.
"The usual $\infty$" is a property that $\omega$ has but it is meaningless to define any ordinal as being $\infty$. All infinite ordinals have this property, but they are not "$\infty$" 
When talking about numbers tending towards infinity we think of numbers getting infinitely big. When talking about ordinals we can think of them as numbers but never do they "become infinity".
