I am busy latedays with "Logic and Structure" of Van Dalen in order to get some familiarity with logic.
Proposition symbols are introduced (on page 7) as members of an infinitely countable set $\{p_0,p_1,p_2,\dots\}$.
My first thought was: why not just a set without any further specification?
Uptil now I could not find a satisfying answer.
Let me mention that I did find a proof (on page 44) that every (syntactically) consistent set of propositions is contained in a set of propositions that is maximally consistent, and in that proof it is used that the set of proposition symbols is countable.
That however can also be proved if there is no countability, but then by application of the lemma of Zorn.
So I wondered if the countability is there because this (i.e. Zorn/axiom of choice) must be avoided for some reason.
I really don't know...
So my question is:
Is there a special reason/motivation to go for an (infinitely) countable set of proposition symbols?
While I have not seen this book, it seems to be an undergraduate text. The statements in propositional logic are finite strings of symbols. So that one does not run out of proposition symbols when writing complicated statements, they allow infinitely many proposition symbols. A proof is a finite list of statements. But then any proof will require finitely many proposition symbols, but there is no upper limit on this finite number. This shows countably many is enough.
There are languages that allow propositions to be infinite strings. There is a Wikipedia page on infinitary logic