TITLE: What do models where the CH is false look like?
QUESTION [6 upvotes]: Additionally, is there any intuitive way to visualize the cardinalities that result?

REPLY [5 votes]: I have an objection to the way in which the question and some of the answers are phrased.  To me it seems meaningless to ask whether the Continuum Hypothesis is "true" or "false".  It is of course meaningful to ask whether CH is true or false in a given model. 
But unless you can reveal to me some objective world of sets, I'll have no idea what you mean by CH being "true" or "false" in an absolute sense.  
It's like asking whether the parallel postulate is "true" or "false".  There are geometrical systems in which it's true, and geometrical systems in which it's false; end of story.  Similarly, we know that from any model of ZFC, one can build (i) another model of ZFC in which CH is true, and (ii) another model of ZFC in which CH is false. 
I know what the original question is asking, but I'm surprised that so many people continue to use this language.  The same goes for the Axiom of Choice; it amazes me when people argue about whether it's "true" or "false".