Cardinality of the set of terms of the Language of Abelian Groups

Question

I believe the cardinality of the set of terms of the Language of Abelian Groups ($\lbrace \mathbb{Z}, 0, +, - \rbrace$) is $\aleph_o$ because $\mathbb{Z}$ is countable and we're performing addition and subtraction on it, so I believe the cardinality should also be countable. Is my reasoning correct. How would you rigorously prove it?

Answer 1

I'll denote with $\mathscr{L}=\mathbb{Z}\cup \{+,-,0\}$ and with $\mathscr{T}$ the set of terms in this language. Clearly:

$$\mathscr{T}\subset \bigcup_{n\in\mathbb{N}_0}\mathscr{L}^n$$

The finite cartesian product of countable sets is countable and the union of a countable family of countable sets is countable. This means that $\mathscr{T}$ is also countable.