Help me understand decidability of an axiom system of a theory

The author of the book I'm reading defined an axiom system $X$ of a theory $T$ like this:

Let $X$ be a set of formulas. Then $X$ is an axiom system of a theory $T$ of which it holds that $T = \{ \alpha \in \mathcal{L}^0 : X^g \models \alpha \}$, where $X^g$ is the universal closure of $X$ and $\mathcal{L}^0$ is the set of sentences of the language $\mathcal{L}$.

He then goes on to define a recursively axiomatizable theory:

A theory is recursively axiomatizable if its axiom system is decidable.

The problem is that there are no clear examples of decidable axiom systems given in my book (thus far). I know that a set $Z$ of strings of an alphabet $\mathsf{A}$ is decidable iff there is an algorithm which tells us after finitely many steps whether a string $\xi$ belongs to $Z$, for all strings $\xi$ which we can form from $\mathsf{A}$.

I think the essence of my problem is that I can't imagine the condition for a formula belonging to $X$ which the algorithm would be checking. I would appreciate any examples and clarifications which would make the concept more clear to me. Thank you.

You basically have all the ingredients: You have a set $X$ of statements as generated by some language $L$. Now, $X$ is decidable if and only if there is an algorithm that can decide whether or not a given string, as generated by that same language $L$, is an element of $X$ or not.

Example: Peano Arithmetic

So here you have 6 axioms plus one axiom scheme, and it is easy to check if any given sentence is one of the 6 axioms o an instance of that axiom scheme. Thus, this is a decidable axiom set.