[FOM] Woodin's pair of articles on CH
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Tue Jan 12 09:39:23 EST 2010
I am trying to work through Woodin's two articles on CH that appeared a
few years ago in the Not. AMS (*after* my subscription lapsed, which is
why i have only just encountered them). I can't hep wondering what
proportion of the readership of the Notices will understand any
significant part of them, but that is by-the by.
W makes the point that $H_{\aleph_0}$ (the sets of sets hereditarily of
size less than $\aleph_0$) is the same size as the naturals and therefore
if you are Mr Magoo you can pretend they are the same thing. In fact if
you are Mr. Ackermann you know the clever binary relation on
$H_{\aleph_0}$ that makes it iso to the naturals.
The next move is to observe that $H_{\aleph_1}$ (the sets of sets
hereditarily of size less than $\aleph_1$) is the same size as the reals.
I know a proof of this fact, but it relies on the fact that there are
precisely continuum many countable sets of reals. I know of no proof of
this equality that does not use (a little bit of) AC. Is there in fact a
proof that doesn't used choice? (My guess is not) and my second question
is: how much does this matter?
tf
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