[FOM] Arithmetic-free theory of formal systems?

[Timothy Y. Chow]

While thinking about how to make the arguments in my previous article
"Indispensability of the natural numbers" more precise, I found myself
coming back to an old question.  Namely, is there a way of developing a
theory of formal systems without any reference to arithmetic?

I think I first saw this idea mentioned in

  Auerbach, David, 1992, "How to say things with formalisms", in  Proof,
  Logic, and Formalization, Michael Detlefsen, ed., London: Routledge,
  77-93.

In the traditional development of the subject, assertions such as "PA is
consistent" are formalized as *arithmetic* statements.  Presumably one of
the motivations for this move is the presupposition that our concept of a
natural number is so simple and clear, that one gains in clarity and
precision by making the translation from syntactic entities to arithmetic
entities.

However, there seems to be a persistent group of people who apparently
find syntactic concepts such as "symbol," "string," "rule," "consistency,"
and so on to be *more* perspicuous than arithmetic concepts such as
"natural number," "successor," "multiplication," and so forth.  Moreover,
there are those who worry that the standard translation of (say) the word
"consistent" into an arithmetic predicate is not a faithful one (Detlefsen
seems to be an example).

This makes me wonder whether it's possible to define a formal language,
dubbed "Syntax" by Auerbach, that directly formalizes the concepts of
"symbol," "concatenation," etc. without reference to arithmetic concepts.
Then one might be able to analyze just how strong an assumption one is
making when one asserts, for example, that "Syntax is adequately captured
by the first-order language of arithmetic."

Since I have trouble grasping what might be lost in the translation from
Syntax to the language of arithmetic, I don't think I'd be the right
person to carry out this job, but perhaps those who see a clear issue
here could carry out such a project (if it hasn't been done already)?

Tim