Sets and the continuum hypothesis

Continuing this post, from this Nature article, which feeds my thesis:

"[...] a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. [...It] efficiently boils down to a question in the theory of sets [...Cantor] was not able to prove this continuum hypothesis, and nor were many mathematicians and logicians who followed him."

"Gödel [...] showed that the continuum hypothesis cannot be proved either true or false starting from the standard axioms — the statements taken to be true — of the theory of sets, which are commonly taken as the foundation for all of mathematics. Gödel and Cohen’s work on the continuum hypothesis implies that there can exist parallel mathematical universes that are both compatible with standard mathematics — one in which the continuum hypothesis is added to the standard axioms and therefore declared to be true, and another in which it is declared false."