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

William Tait wwtx at earthlink.net
Tue May 18 13:50:28 EDT 2004

On May 17, 2004, at 5:35 PM, Timothy Y. Chow wrote:

> 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?

I THINK that the reference to Quine's Mathematical Logic (the Chapter 
on Syntax) is relevant to your question---but you must read through (or 
not) a lot of informal discussion.
Bill Tait

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