To answer your first question:  Yes.  I mean, both.  (And no, I'm not offended by your choice of words.)<p>Since you've restated the discussion on more neutral terms, allow me to rephrase my position in your terms:  The article is about how people have trouble swallowing Cantor's argument.  Yet clearly Cantor's argument is correct within a framework that includes the reals.  And my position is that the existence of Cantor's argument leads me to question the framework itself.<p>You say that there really isn't anything to agree or disagree with, that we are just talking about definitions.  Absolutely, I agree.  It's unfortunate that the only words we have to talk about these things also have extra connotations that might be more forceful than what we really mean.  We're really just talking about the consequences of manipulating symbols under certain rules.<p>HOWEVER...  Isn't the whole point of math that you expect at some point to analyze the result of the symbol manipulation and "read off" an interpretable meaning from the answer?  At that point may you not make a judgement that the combination of the system you choose and your interpretation of it lead to "believable" results?  Is it entirely correct to say that there is no such thing as "agreeing" or "disagreeing" with a definition of a mathematical object, if putting it into the system leads to results you don't consider believable?