On the existence of certain total recursive functions in nontrivial   axiom systems, I
  We investigate the existence of a class of ZFC-provably total recursive unary functions, given certain constraints, and apply some of those results to show that, for $\Sigma_1$-sound set theory, ZFC$\not\vdash P<NP$. 
N. C. A. da Costa (University of Sao Paulo) and F. A. Doria (Federal   University at Rio de Janeiro)