Cardinality of a language $L_\Sigma$ over a decidable signature $\Sigma$

In the middle of a proof of a theorem I was studying, in order to prove a cardinality argument, there was the following statement:

Note that $|L_\Sigma|=|\Sigma|+ \aleph_0$

Where $L_\Sigma$ is a language over a decidable signature $\Sigma$. Intuitively I think it makes sense because the signature is decidable and all, but rigorously why is this so? (Feel free to ask for any clarification on notation or definitions if they are ambiguous, the book for this course often uses unusual names or formulations). Thanks in advance.

Solution 1:

Decidability is irrelevant for cardinality. It is always true tahat $|L_\Sigma|=|\Sigma|+ \aleph_0$. Infact $L_\Sigma\subseteq (\Sigma\cup C\cup V)^{<\omega}$ where $C$ is the set of logical connectives and $V$ is a set of varables, which we may assume countable.

I understand $L_\Sigma$ is the set of formulas with extra logical symbols in $\Sigma$. I.e. a set of finite sequences of symbols in $\Sigma\cup C\cup V$.