The actual meaning of "continuum hypothesis can’t be proved one way or the other" is that the axioms we use in set theory are not sufficient to uniquely define

The actual meaning of "continuum hypothesis can’t be proved one way or the other" is that the axioms we use in set theory are not sufficient to uniquely define what set is. They allow multiple possible interpretations. Under some of those interpretations, an infinity between N and R exists, under other interpretations it does not. As a simple example, if I make the following set of axioms -- "each natural number is either big or small", "10 is a small number", "20 is a big number", "it cannot be simultaneously true that X is big and X+1 is small", -- then I can prove that 5 is small, or that 30 is big, but I can neither prove nor disprove that 15 is big. Because this set of axioms allow multiple possibilities for what "big" and "small" mean.