# [FOM] Inconsistency of P

Monroe Eskew meskew at math.uci.edu
Thu Oct 6 15:56:38 EDT 2011

Here is some further explanation of what motivated the discussion.
Please correct me if I say anything wrong:

The following claim of Raatikainen suggests a way to adjust Nelson's argument.

Claim: "If a subtheory would prove K(n)>c, it is not necessarily
inconsistent, but then... it must fail to be Sigma_1 complete."

If T is sufficiently strong, then T can define Delta_0 truth and prove
the following "internal" version of Sigma_1 completeness:

For all Delta_0 formulas \phi(x), if there is n such that "\phi(n)" is
true, then Q proves "\exists x \phi(x)."

Hence we might use Raatikainen's claim to adjust the argument as follows:

If S proves K(n) > c(T), then S is inconsistent or Sigma_1 incomplete.
Now T proves S is consistent and Sigma_1 complete.
So T proves S does not prove K(n) > c(T).
...

This is why I was interested in refuting the claim of Raatikainen.

-Monroe