[FOM] Inconsistency of P
dmehkeri at gmail.com
Wed Oct 5 21:01:13 EDT 2011
> 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).
Yeah, I imagine other FOM readers are also wondering why we care at all.
One place where it would matter, even if T is consistent, is when T is
reasoning about its own subtheories. The fact that T is consistent is
not available within T. As I understand it, this was relevant to the
If S proves K(n) > c, then S is inconsistent.
[BUT we can only say T is inconsistent]
Now T proves S is consistent.
So T proves S does not prove K(n) > c(T).
So T is inconsistent [by something like Kritchman-Raz's unexpected
But the computation of this inconsistency is superexponential.
Now PRA proves superexponential functions terminate.
But PRA also proves T is consistent.
So PRA is inconsistent.
This is also a possible answer to Taylor's question about what it might
mean to find an infeasible inconsistency.
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