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.

翻译：在过去50年间，可计算性领域中许多问题惊人地未获解答。例如，$P$与$NP\cap co-NP$关系的问题。这一问题的通俗表述是："如果一个人有能力验证某个问题的证明和反证，那么这个人是否知道该问题的解？"谈及人类时，显然可以看到该问题取决于具体个体对问题的认知程度。我们的主要目标是将这一观察扩展到集合论形式模型$ZFC$：给定一个模型$M$和$NP\cap co-NP$中的特定问题$L$，如果我们"认知"问题$L$，则可以证明$L$属于$P$。本文首先定义认知概念，并阐释其为何符合直观的认知概念。随后构建一个我们能够认知众多函数的模型。基于该模型的存在性，我们推得任何包含世基基数（worldly cardinal）的模型中，我们都能认知一类广泛的函数。由此证明：若假设存在世基基数，则"某个可证明属于$NP\cap co-NP$的可定义语言同样属于$P$"这一命题是可证的。在假设世基基数存在的前提下，我们通过简单运用这些定理，即可在多项式对数时间内完成整数分解。本文虽未解决$P$与$NP\cap co-NP$的关系问题，但其主要结果使我们向判定该问题迈近了一步。