This is wild conjecture on my part; just intuition, but "both proving as well as disproving the continuum hypothesis is impossible within the current axiomatic system" suggests a portal to an unknown dimension. Mathematicians have developed a powerful sytem that mirrors our reality on an abstract yet practical level; counting, calculus, probability theory etc. However, when something is shown to be neither proveable nor disproveable it reminds me of Schrödinger's cat and the paradoxes of the quantum world. When our mathematical axioms fail a portal to a new reality opens before us.