computer science – Is the set of all regex that match themselves countably infinite?

With my meager computability knowledge, I believe that this question is equivalent to “Is the problem of enumerating all regex that match themselves decidable?” or maybe some variant on that idea.

And of course, by regex I mean regular expression in their simple, stack-less form.

As an example, the regex a will match itself.
Conversely, the regex [a] will not, because it will look for a as the only possible first character and not find it.
Going further, .*[a]? matches itself because the .* will resolve to the entire regex and the [a]? will resolve to nothing.