[FOM] Simple historical Question: re: Jones, Heck

Allen Hazen allenph at unimelb.edu.au
Thu Jul 26 02:21:38 EDT 2007

Richard Heck wrote:
>Roger Bishop Jones wrote:
>>  However, a vigorous defence was mounted.
>>  If A is moderated to "arithmetic truth is not arithmetically definable",
>>  then the argument is sound, and can be generalised to give stronger
>>  negative results than that arithmetic truth is not recursively enumerable.
>I heard Quine lecture on this very point at BU in, oh, it must have been
>1994 or so. His point was precisely that Tarski's theorem, stated in
>that form, is stronger than G"odel's.

---And Quine is an appropriate person to point 
that out! The final chapter ("Syntax") of his 
1940 book "Mathematical Logic" is a very nice 
exposition of Gödelian/Tarskian themes, using his 
language of "protosyntax" (= First-Order theory 
of strings of symbols) to excellent pedagogical 
effect.  (Many early readers of Gödel were 
puzzled by all the coding and "translation' -- 
one article about it in the philosophical journal 
"Mind" was entitled "Through Babel to Gödel" -- 
and by using a language that is explicitly about 
formal objects for part of the proof Quine 
managed to reduce the mental strain on the 
reader.)  What's relevant here is his final 
result in the chapter: modulo the 
interinterpretability of Protosyntax and 
First-Order Arithmetic (proven partly in the 
chapter and partly in Quine's 1956 JSL article 
(repr. in his "Selected Logic 
Papers")"Concatenation as a basis for 
arithmetic"), it's that the set of truths of 
"logic" -- here interpreted as including set 
theory-- is not arithmetically definable.  So, 
though the idea would not have come as a surprise 
to Gödel or Tarski, Quine was perhaps the first 
to publish a widely accessible proof of this 
version of the theorem.


Allen Hazen
Philosophy Department
University of Melbourne

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