These are the definitions I'll be working with. (Note that they are informal and the author promises more formal ones later on in the book, but I'm not there yet.)
A set $Z$ of strings of a given alphabet $\mathsf{A}$ is called decidable if there is an algorithm (a mechanical decision procedure) that after finitely many calculation steps provides us with an answer to the question whether a string $\xi$ of symbols of $\mathsf{A}$ belongs to $Z$. [...] A theory $T$ is called recursively axiomatizable, or just axiomatizable, if it possesses a decidable axiom system.
At first glance, it seems to me that an uncountable theory $T$ could be decidable, as it seems possible that an algorithm could determine whether a string belongs to its axiom system $X$, for all strings of an alphabet. However, here my doubts come up. The author provides a sketch of the following theorem.
The theorems of an axiomatizable theory are effectively enumerable.
He defines enumerability in this way.
Put roughly, a set $M$ [...] is called effectively (or recursively) enumerable if there exists an algorithm that delivers the elements of $M$ stepwise.
However, if an axiom system $X$ of some theory $T$ is uncountable, so is $T$ itself, and there can be no algorithm that could deliver the elements of $T$ (and therefore of the set containing all of the theorems of $T$, since $T$ is a subset of this set) stepwise. (This is because there is no bijection $f\colon \, \mathbb{N} \to T$, as $T$ is uncountable.) The conclusion seems to be that $T$ being axiomatizable implies it being countable.
My question concerns my understanding of the definition of decidability. I interpreted the author's definition as saying that the algorithm deciding the membership of each string $\xi$ runs on all strings simultaneously. If the author meant to say that the algorithm can only process one string per step, then it, of course, trivially follows that no uncountable set is decidable. I am grateful for all clarifications of this confusion. :)