Unlike a natural, or even a real number, sets are built in a transfinite hierarchical manner as our ability to conceptualize them. Difficulties begin to appear around unambiguously conceiving uncountable cardinalites. What I consider important to remember is that sets are objects of thought,therefore we can say that a sentence is true when we find sufficient reason for it to be true . That means, what we know is true is what we can Prove is true, for some unfinished yet ever-expanding notion of Proof. Now , the crucial point to remember is how the 'Proof modality' works .
All laws hold except the excluded middle .It isn't necessary either to Prove a or to Prove not a . While the logic of ontology is aristotelian , the logic of epistemiology (what we can prove) is intuitionistic . In Peano arithmetic we have a 'rule of thumb' : If you cannot prove (exists x , G x), for a simple enough sentence G , then (for all x , not(G x)) may be considered provable . An extension of this principle for set theory would be wonderful . Although , while PA seems to obey a 'minimality principle' , set theory seems more suited to a 'maximality principle' . Given that epistemology is intuitionistic, I'm open to the possibility that neither CH nor its negation could provide a 'final argument' .