[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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