[FOM] Is Godel's Theorem surprising?
mario
chiari.hm at flashnet.it
Mon Dec 11 12:41:32 EST 2006
Hi,
I guess the reference is the following one (thanks to Google power,
search string `Godel Kripke Putnam´)
http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1027953483
A remark: the logical complexity of the (new) indipendent proposition
seems to be \Pi^0_2; if so, the result would be weaker than Godel`s.
Warren Goldfarb has a very interesting paper which discuss a similar -
or even the same - idea by Kripke:
Warren Goldfarb, Herbrand`s Theorem and the Incompleteness of
Arithmetic, Iyyun, A Jerusalem Philosophical Quarterly 39 (Jan 1990)
(Iyyun info page: http://socrates.huji.ac.il/iyyunen.html)
Goldfarb comments on incompleteness and self-reference too.
I have a somehow related historical question: who was the first to guess
that formulations of ordinal induction and recursion could be used as a
threshold for provability within a formal system? In particular, is
there any evidence that Ackermann had some idea that its own definitions
by ordinal recursion could be used to provide such kind of threshold?
my best wishes
mario
On Sun, 2006-12-10 at 15:53 -0500, Harvey Friedman wrote:
> On 12/10/06 9:19 AM, "Charles Silver" <silver_1 at mindspring.com> wrote:
>
> > First, thanks very much for all the interesting and enlightening
> > responses to my question. A couple of comments:
> > Diagonalization is not central to Godel's (first) theorem, as shown
> > by Kripke's proof of G's theorem that was published by Putnam, which
> > does not *require* diagonalization.
> > I believe this proof also shows--please correct me if I'm wrong--
> > that a specifically *mathematical* proposition (though an unusual
> > one) cannot be proved nor can its negation.
> >
> >
>
> It would be helpful to the FOM readership for you to give us a reference to
> this paper by Putnam. I have serious doubts about the claims you are
> suggesting.
>
> Harvey Friedman
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