2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59391We formulate the $P<NP$ hypothesis in the case of the satisfiability problem as a $Π^0_2$ 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 ${\cal T}$.LaTeX, 19 pages, no figuresLogictext