[FOM] Inconsistency of P

Daniel Mehkeri 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
> construction

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?

Daniel Mehkeri

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