[FOM] Arithmetic-free theory of formal systems?

[William Tait]

On May 17, 2004, at 3:37 PM, 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?
>
>> that directly formalizes the concepts of "symbol,"
>> "concatenation," etc. without reference to arithmetic concepts.
>
> One good example is:  Jeroslow, R. G., "Redundancies in the
> Hilbert-Bernays derivability conditions for Gdel's second 
> incompleteness
> theorem", J. Symbolic Logic 38 (1973), 359-367.
>
> There's also Smullyan's "Theory of Formal Systems", though I found it
> harder to read.  --Matt

Smullyan's book is well worth the effort. But, in direct answer to 
Timothy Chow's question, Quine (Mathematical Logic Chapter 7, entitled 
``Syntax'' develops a theory of concatenation of symbols up to (at 
least) a proof of the first incompleteness theorem.

Bill Tait
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