[FOM] Explanation/Continuum Hypothesis
Timothy Y. Chow
tchow at alum.mit.edu
Tue May 9 14:59:33 EDT 2006
"Studtmann, Paul" <pastudtmann at davidson.edu> wrote:
> Tim Chow wrote
>
> >Paul Cohen, in his book "Set Theory and the Continuum Hypothesis," said
> >something to the effect that the powerset axiom is a bold new way of
> >constructing sets and that one should not expect to be able to reach the
> >powerset from below in piecemeal fashion.
>
> Can you explain a bit more as to how this counts as an explanation for
> one half of the independence of the continuum hypothesis. Why is the
> powerset axiom so bold and what makes a powerset somehow unreachable
> from below in a piecemeal fashion? And what precisely is the connection
> between being unreachable from below in a piecemeal fashion and failing
> to be a certain size?
FOMers may be confused because you're quoting from an email I sent to you
off-list. So let me first repeat what I said in that email: Cohen really
intended this to be an explanation for why CH is *false* (though perhaps
it's not unreasonable to co-opt them into an explanation for why CH is
*unprovable*). He also later moved away from this point of view to a more
"formalist" stance.
I don't think Cohen gave much more explanation that what I have sketched,
so to try to answer your question, I have to inject my own speculations as
to what Cohen was thinking. The "mundane" methods of getting bigger
cardinals are taking successors and taking unions. The powerset axiom is
bold because it postulates an entirely new method of creating sets; we
can't, for example, derive the powerset axiom from the other axioms. Thus
there is no reason to expect that we can get all the way to 2^(alpha_0) by
mundane methods of getting bigger cardinals, let alone by taking a single
step forwards from aleph_0.
Tim
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