  We formulate the $P<NP$ hypothesis in the case of the satisfiability problem
as a $\Pi ^0_2$ sentence, out of which we can construct a partial recursive
function $f_{\neg A}$ so that $f_{\neg A}$ is total if and only if $P < NP$. We
then show that if $f_{\neg A}$ is total, then it isn't ${\cal T}$--provably
total (where ${\cal T}$ is a fragment of ZFC that adequately extends PA and
whose consistency is of ordinal order). Follows that the negation of $P < NP$,
that is, $P = NP$, is consistent with those ${\cal T}$.

Logic