[FOM] Re: Arithmetic-free theory of formal systems?

[Timothy Y. Chow]

On Mon, 17 May 2004, Matthew Frank wrote:
> On Mon, 17 May 2004, Timothy Y. Chow wrote:
> > is there a way of developing a theory of formal systems without
> > any reference to arithmetic?
[...]
> There's also Smullyan's "Theory of Formal Systems", though I found it
> harder to read.  --Matt

Thanks for the suggestions (by you and by those who responded to you).
Before I go look them all up, however, perhaps I should clarify exactly
what I was asking for.

I want something one level more formal than Smullyan's book.  Smullyan
defines formal systems using English sentences such as "An alphabet is
a finite set of elements called symbols" and "Given any n symbols, ..."
I am looking for a *formal* language whose models are syntactic entities.
To get this out of Smullyan's book, I would have to formalize his informal
treatment.  But if I do so in the most obvious way, I run into the words
"finite set" and "n" in the English sentences quoted above, and then I
find myself driven to formalize fragments of set theory and arithmetic,
which is a garden path I don't want to go down if I can avoid it.

Recall that the context of my question was a hypothetical person who is
doubtful about natural numbers but who thinks that symbols and rules and
so forth are perfectly clear concepts.  When he speaks, he eschews as far
as possible any nouns and verbs from arithmetic.  I want to formalize his
patterns of speech, so that I can compare it with formal theories of
arithmetic such as PA, PRA, etc.  The goal is to address his doubts about
whether the natural numbers are coherent and whether arithmetizations of
the concept of consistency adequately capture the original syntactic 
concept.

With this clarification, do the references people have cited still address 
my question?

Tim