[FOM] Inconsistency of P
dmehkeri at gmail.com
Tue Oct 4 22:01:04 EDT 2011
> > If S proves "K(n)>c" for some n, then we can
> > indeed conclude that T is inconsistent. But there need not be any
> > special deficiency of S in consistency, truth, or strength, since c
> > depends on T-- it is c(T), not c(S).
> Well, yes, if by c(S) you mean the value given by the Chaitin machine
Yes, the machine version applies to arithmetisations, and does not need
the hypothesis of consistency (and in the original context, this matters).
> (though, as I emphazides, S is then not a subtheory of T)
I still don't understand why.
Suppose S is a subtheory of T, and S proves K(n) > c(T) for some n.
So T proves K(n) > c(T) for some n;
so the Chaitin machine construction finds m such that K(m) <= c(T) and
such that T proves K(m) > c(T);
so T is inconsistent.
But m might not be n. We might still have c(S) >= K(n) > c(T).
So, S could be a (non-deficient) consistent subtheory of T that proves
K(n) > c(T). Am I still missing something?
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