  Over the course of the last 50 years, many questions in the field of
computability were left surprisingly unanswered. One example is the question of
$P$ vs $NP\cap co-NP$. It could be phrased in loose terms as "If a person has
the ability to verify a proof and a disproof to a problem, does this person
know a solution to that problem?".
  When talking about people, one can of course see that the question depends on
the knowledge the specific person has on this problem. Our main goal will be to
extend this observation to formal models of set theory $ZFC$: given a model $M$
and a specific problem $L$ in $NP\cap co-NP$, we can show that the problem $L$
is in $P$ if we have "knowledge" of $L$.
  In this paper, we'll define the concept of knowledge and elaborate why it
agrees with the intuitive concept of knowledge. Next we will construct a model
in which we have knowledge on many functions. From the existence of that model,
we will deduce that in any model with a worldly cardinal we have knowledge on a
broad class of functions.
  As a result, we show that if we assume a worldly cardinal exists, then the
statement "a given definable language which is provably in $NP\cap co-NP$ is
also in $P$" is provable. Assuming a worldly cardinal, we show by a simple use
of these theorems that one can factor numbers in poly-logarithmic time.
  This article won't solve the $P$ vs $NP\cap co-NP$ question, but its main
result brings us one step closer to deciding that question.
