[xamuel]

Presumably drvd is referring to the independence of the Continuum Hypothesis: we don't know |R| in the sense that we don't know for which ordinal alpha is |R| the alpha'th infinite cardinal. The Continuum Hypothesis is the claim that |R| is the first cardinal after |N|. It's well known the Continuum Hypothesis is independent of ZFC. That means for the question of "how many reals there are" (at least if that question is read in the formal sense), the answer is "it depends: for different models of ZFC, there are different answers".