Numbers Extensions

TL;DR

This paper explores the concept of 'knowledge' within formal set theory models to relate it to computational complexity, proposing that assuming certain large cardinal axioms can imply P=NP∩co-NP and enable fast number factoring.

Contribution

It introduces a formal notion of knowledge in set theory models and links it to complexity classes, showing how large cardinal assumptions influence computational problems.

Findings

Defined a formal concept of knowledge in set theory models.

Constructed models with knowledge of many functions under large cardinal assumptions.

Proved that assuming a worldly cardinal, certain NP∩co-NP problems are in P.

Abstract

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.