From: ikastan@uranus.uucp (Ilias Kastanas)
Subject: Re: Sensible Godel Theorem Question
Date: 1 Sep 2000 01:54:31 GMT
Newsgroups: sci.math
Summary: [missing]
In article <39ACB84E.E1E98FA1@trump.net.au>,
William Hart wrote:
...
>
>What about the statement you give. Suppose we have:
>
>G: G is provable.
>
>This has the property you specify. Is G provable in the formal system?
>
>I can't see that it is. (But then I can't see that it isn't). What do people
>think?
>
>
People didn't quite know what to think... until Loeb settled
the matter. A sentence H such that
T |- H <-> Prov_T('H')
("asserting its own provability") is called a Henkin fixed-point. Any
H is provable in T... and true (a nice counterpart to: any Goedel
fixed-point is unprovable in T, and true). T, and Prov_T, are assumed
to satisfy certain natural hypotheses (that also yield Goedel's theorem).
Other fixed-points have been investigated (Rogers, Jeroslow...);
see Boolos's excellent "The Unprovability of Consistency". And the
situation has been generalized; appropriate names, if you want to do a
search, are: Bernardi, Smorynski, Sambin, De Jongh.
Ilias