I'm reading Chang and Keisler's Model Theory and I don't quite understand the notation they use for the cardinality of a language.
Elsewhere in the book, the cardinality of a set $X$ is denoted by $|X|$. Such as in p.21 where they talk about the cardinality of a model $\mathcal{A}$ being the cardinality $|A|$ of its universe $A$. (I checked, a similar usual notation is found in the Set Theory appendix of the book, where a cardinal definition is to be found).
What I don't understand is the following notation (section 1.3, p.19) of the cardinal of a language $\mathcal{L}$ :
$||\mathcal{L}|| = \omega \cup |\mathcal{L}|$
If the cardinal of a language is $||\mathcal{L}||$, then what is $|\mathcal{L}|$ ?
And why does it have a cardinality of at least $\omega$ ? I guess this has to do number of variables. But then again, what would $|\mathcal{L}|$ be ? It's not clear to me. I guess $|\mathcal{L}|$ could be the cardinality of the set of symbols of $\mathcal{L}$, but isn't that precisely what the cardinality of $\mathcal{L}$ is supposed to be ? In which case, why do they decide to denote it by $||\mathcal{L}||$ ?
As suggested by Arthur Fischer, $\rVert \mathcal L\rVert$ is the number of (first order) formulas in $\mathcal L$.
From my experience, I've seen $\lvert \mathcal L\rvert$ used to denote what is denoted here as $\lVert \mathcal L\rVert$ (i.e. the cardinality of the set of formulas).
This vagueness originates, I think, from simple convenience (as mentioned in the comments) and from the blurred distinction between languages (as sets of formulas) and signatures in first order logic.
It is also kind of similar to how we sometimes denote by $\lvert M\rvert$ the cardinality of the universe of a model $M$, instead of writing $\lVert M\rVert$, which is technically the correct way to denote it – this is because we often denote by $M$ both the model and its universe, which we could denote by $\lvert M\rvert$ if we wanted to be more precise, or else we could denote the model by $\mathfrak M$ and the universe by $M$ – but both of these conventions are bothersome, and confusion they might avert is unlikely, anyway.