[FOM] Inconsistency of P
Panu Raatikainen
panu.raatikainen at helsinki.fi
Wed Oct 5 02:46:42 EDT 2011
"Daniel Mehkeri" <dmehkeri at gmail.com>:
>> (though, as I emphazides, S is then not a subtheory of T)
>
> I still don't understand why.
I think we have now sorted this out with Monroe Eskew in a couple of
off-list e-mails.
Focusing on the content of Chaitin's theorem rather than on any
particular proof of it, I assumed that c is a finite limit such that T
cannot prove K(n) > c.
But that, of course, tacitly presupposes that T is really consistent.
So I thought the inconsistency of T was only hypothetical: "If T
proved K(n)>c, then it would be inconsistent".
I did not take the alternative that T *really* is inconsistent
seriously, because then c, understood as above, would not even be
defined.
* * *
If c, on the other hand, is only a suitable constant which is involved
in a Chaitin machine construction for T (inconsistent), then a
consistent subsystem S of T may prove K(n) > c.
Or, at least, it is not clear to me it could not.
This case just is not very interesting, because *all* consistent
theories (in the language of T) are subsystems of T; and trivially,
some of these prove K(n) > c (let c be however large and complex).
Best
Panu
--
Panu Raatikainen
Ph.D., University Lecturer
Docent in Theoretical Philosophy
Theoretical Philosophy
Department of Philosophy, History, Culture and Art Studies
P.O. Box 24 (Unioninkatu 38 A)
FIN-00014 University of Helsinki
Finland
E-mail: panu.raatikainen at helsinki.fi
http://www.mv.helsinki.fi/home/praatika/
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