The answer by jmite tells you what is being done in computer science.
The point of it is that it is primarily devoted to algorithm and
computation, as ways of solving problems. Hence it is also tightly
connected with proof theory, and the logical work in the late 19th and
early 20th century.
To a large extent, it is a theory of syntax, as all we do in
mathematics is to push symbols around according to specific rule. Of
course, as you know, there is semantics behind the symbols and their
assembly into sentences, but the only way to refer to the semantics is
through the symbols.
In a way, it is a physical theory, as much as a mathematical one. It
is about what we can express in this physical world, as human being,
or using machines. It is about computation, which is also called
calculus (not to be confused with differential and integral calculus),
which comes from counting stones in Latin. Our first interest is then
to develop what seems physically meaningful.
Of course, there are probably many ways to extend it, such as
hypothesizing computable solutions to problems even though none actually
exists, which is somewhat like adding an axiom that has no physical
model. It is a bit like science-fiction: we can talk about the extension, and what it can do , and how that would change the world. But we cannot actually use it and do those things.
So, nearly everything we do in practice is finite, but often cannot be
bounded meaningfully. Working up to denumerable infinity turned out to
be the most convenient way to address efficiently that situation. We
often consider "infinite" structures, but in the end they are only
limits of uniformly computable finite approximations, which keeps us
in the denumerable world.
The case of computable reals is interesting. They are (isomorphic to)
a subset of the usual reals, but defined as computable limits, hence
characterized by whatever machine computes the approximations. In the
end, each is characterised by a finite definition, which makes them
denumerable. The interesting point is that you can do with computable
reals practically all you do with the classical ones. See for example "To what extent can the mathematics of Reals be applied to Computable Reals?"
My guess, but it is only a guess, is that the same is probably true of
infinitiary logic (of which I know only what you said). If you are
careful to consider only computable strings on infinite length, you
will stay in the denumerable realm, but you can probably adapt all
definitions to get whatever nice properties you expect from
infinitiary logic. And, as a bonus, it may have physical sense.