[FOM] Explanation/Continuum Hypothesis

[Studtmann, Paul]

Some time ago there was a thread on FOM concerning the question of
explanation in mathematics. I have a series of questions concerning this
matter that are directed at a particular case, namely the independence of
the continuum hypothesis.  I am curious to know the opinions of
mathematicians and/or philosophers about the following.
 
1. Is there some explanation for the independence of the continuum
hypothesis from the other axioms of ZFC that goes beyond the claim that
neither it nor its denial is entailed by the other axioms of ZFC.  [For
instance, I once read in a book on set-theory that the continuum hypothesis
is independent from the other axioms because we do not have a clear
conception of the nature of infinite sets.  I am not suggesting that this is
a good explanation.]
2. If the answer to (1) is Œyes¹, what is the explanation?
3. If the answer to (1) is Œno¹, why not?
4. Are there well-known schools of thought with respect to this question, in
the same way that there are, for instance, well-known schools of thought
about mathematical objects ­ the Platonists, structuralists, conceptualists,
nominalists etc. 
5. If the answer to (4) is Œyes¹, what are the schools?
6. If the answer to (4) is Œno¹, are there any suggestions as to natural
groupings of views that might emerge concerning this issue?
 
 
 
Paul Studtmann