[FOM] Inconsistency of P

Rempe, Lasse L.Rempe at liverpool.ac.uk
Tue Oct 4 06:54:07 EDT 2011

> Lainaus "Monroe Eskew" <meskew at math.uci.edu>:

> > No, in my specific counterexample, I showed a situation in which
> > "K(n)>c" is actually a TRUE sentence, proven by a consistent S
> > extending PA, where (paradoxically?) c is actually the constant coming
> > from a Chaitin machine for a strictly larger theory T (the axioms of T
> > are a superset of the S axioms), where T is inconsistent.
> Look, forget witnesses, forget Chaitin machines, forget any particular  
> way to demonstrate Chaitin's theorem!
> In the end of the day, what Chaitin's theorem says that given a theory  
> T, there is a constant c such that T cannot prove "K(n)>c" for *any*  
> natural number. [1]

The word "consistent" seems to be missing (or implicitly assumed) in the preceding sentence. In the trivial example mentioned by Monroe Eskew, the theory T is inconsistent, but for every number c there is a consistent subtheory that proves K(n)>c for some n. 

There does not seem to be any inconsistency (pardon the pun) between the two points of view.

(I am, however, not entirely sure which specific number c Monroe had in mind, given that the theorem obviously does not apply to inconsistent theories).

More information about the FOM mailing list