A lemma on a total function defined over the Baker-Gill-Solovay set of polynomial Turing machines

If we establish that the counterexample function for P=NP, if total, overtakes all total recursive functions when extended over all Turing machines, then what happens to the same counterexample function when defined over the so-called Baker-Gill-Solovay (BGS) set of poly machines? We state and prove here a lemma that tries to answer this query.Comment: LaTe

On a total function which overtakes all total recursive functions

This paper discusses a function that is frequently presented as a simile or look-alike of the so-called ``counterexample function to P=NP,'' that is, the function that collects all first instances of a problem in NP where a poly machine incorrectly `guesses' about the instance. We state and give in full detail a crucial result on the computation of Goedel numbers for some families of poly machines.Comment: LaTe

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

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 ${\cal T}$.Comment: LaTeX, 19 pages, no figure