Axiomatic set theory beyond the continuum hypothesis

Abstract

There exist mathematical statements that can neither be proven nor disproven, collectively referred to as independent statements. This phenomenon is fundamentally different from problems that have yet to be solved, as independent statements can never be decided to be true or false. To observe what exactly causes this would require an understanding of the underlying axiomatic system and assumptions that one is working with. It is thus a meaningful mathematical endeavour to study these statements in detail, and gain insights into what makes them independent.

Of course, there are also debates as to whether such independent statements should be true or false in the first place. These arguments fall into the realm of philosophy, and considerable efforts have been made by many great minds in the past to justify both the case for and against these theorems. What we are left with at this point today are the ZFC axioms. The study of set theory and independent statements in this report will however, break free from the shackles of ZFC and look towards stronger hypotheses and extensions of our set-theoretic universe.