On the consistency of $P=NP$ with fragments of ZFC whose own consistency strength can be measured by an ordinal assignment

Abstract

We formulate the $P<NP$ hypothesis in the case of the satisfiability problem as a ${\Pi }_2^0$ 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 $\mathcal{T}$--provably total (where $\mathcal{T}$ 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 $\mathcal{T}$.

LaTeX, 19 pages, no figures