The definition of the cardinality of a signature Here's the definition of the cardinality of a signature. Can anyone elaborate on what "the least infinite cardinal >= the number of symbols in L" mean? What is the cardinality of a signature in the end? This is from the book Model Theory by Wilfrid Hodge

 A: I should remark that the following terminology varies with authors, for example, in Chang and Keisler's Model Theory, the term 'signature' is never used. Some authors  do not take into account the logical vocabulary, because they take it already fixed across languages, etc. However, I find it a clearer practice to distinctly incorporate the terms all, which is quite helpful outside mathematics as well.
The cardinality of a (formal) language $\mathcal{L}$ is effectively the cardinality of the set of wffs (well-formed formulas) expressible in the language $\mathcal{L}$. For example, the cardinality of the language of the standard first-order predicate logic is countably infinite. I shall expand on this below.
The cardinality of a signature $\sigma$ (the non-logical vocabulary of a language $\mathcal{L}$) is the cardinality of the set of symbols in the signature $\sigma$. For example, the signature of the structure $\mathcal{N}=\langle\mathbb{N}, 0, 1, +, \cdot, <\rangle$ has a cardinality of 5.
The cardinality of a structure is equal to the cardinality of its domain (universe). Thus, the structure of the foregoing example has the cardinality of $\vert\mathbb{N}\vert = \aleph_{0}$.
Returning to Hodges' definition of the cardinality of a language:

the cardinality of $\mathcal{L}$, in symbols $\vert\mathcal{L}\vert$
is the least infinite cardinal $≥$ the number of symbols in $\mathcal{L}$

In Chang and Keisler's Model Theory (p. 19), the cardinality of a language is formulated as
$$\Vert\mathcal{L}\Vert = \omega\cup\vert\mathcal{L}\vert,$$
in which $\mathcal{L}$ is substituted for the present usage of signature, employing a sum of ordinal numbers, which might be more informative and useful.
Skipping over the shortcut terminologies of both books (one reason Chang and Keisler's book resorts to the symbol $\Vert .\Vert$, see the question), an explicit formulation of the idea is
$$\vert\mathcal{L}\vert = max(\vert\sigma\vert, \aleph_{0})$$
Notice that the former also one fulfils the same idea: The cardinality of the language is that of which is the larger. We may make sense of the occurrence of $\aleph_{0}$ (or $\omega$) considering a 1-1 correspondence of each wff to a natural number when the signature is finite.
