[FOM] Simple historical question

Roger Bishop Jones rbj at rbjones.com
Sun Jul 22 04:20:32 EDT 2007

On Friday 20 July 2007 19:09, H. Enderton wrote:

> As for your pedagogical point, I quite agree.  Now that we have a robust
> theory of computability, I think we can say that the heart of Goedel's
> first incompleteness theorem lies in the fact that true arithmetic is
> not computably enumerable.  Of course, in 1931 the computability
> concepts were unavailable.

We had a thread about this a while ago "Disproving Godel's explanation of 
incompleteness" when I read that:

> > Godel believed that:
> >
> > (A).  The truth predicate of a language cannot be defined in
> > that language.
> >
> > and
> >
> > (B).  That (A) explains the incompleteness of arithmetic.

My inclination, contra Goedel, was that "the true reason" is more plausibly 
that arithmetic truth is not recursively enumerable, and that his argument 
was unsound because premise A is false.

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.

Apparently Godel's discovery of the theorem came via an anticiparion of 
Tarski's result on definability, and the publication as an incompleteness 
result happened because Godel wanted to avoid a result which involved 
semantics (a new and controversial topic).

Roger Jones

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