[FOM] Woodin's pair of articles on CH

[William Tait]

On Jan 12, 2010, at 8:39 AM, Thomas Forster wrote:
> 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?

What is wanted is to see that each of second-order number theory NT^2 and H(aleph_1) can be interpreted in the other. That NT^2 is interpretable in H(aleph_1) is obvious: take the natural numbers to be von Neumann ordinals and then all the objects of NT^2 are in H(aleph_1). In the other direction, a hereditarily countable set can be coded (not necessarily uniquely) by a well-founded tree whose nodes are natural numbers, which in turn can be coded by a set of natural numbers.  The relation between two trees representing the same hereditarily countable set is definable in NT^2.

I may be misremembering, but I think that Steve Simpson wrote a paper bearing on this in one of the past centuries.  

Bill Tait