# [FOM] Woodin's pair of articles on CH

William Tait williamtait at mac.com
Sat Jan 16 15:26:48 EST 2010

On Jan 16, 2010, at 12:56 PM, Monroe Eskew wrote:

> On Fri, Jan 15, 2010 at 5:26 AM, William Tait <williamtait at mac.com> wrote:
>>
>> I don't see the problem: interpretations needn't preserve identity.  The identity relation between HC sets becomes an equivalence relation between the well-founded trees. That relation, as I pointed out, is definable in NT^2.
>>
>> Bill
>>
>
> There must be some problem because of the models in which the
> proposition fails pointed out by Andreas Blass.
>
> Mapping to the equivalence class of well founded partial orders on
> omega avoids choice at that step.  But the equivalence classes are not
> in H_{\omega_1} since they are of size continuum.

Excuse me for not being clearer. Thomas Forster originally took Woodin's statement that the structure <H(omega_1), \in> is "essentially just the structure  < P(N), N, +, ., \in> to imply that  H(omega_1) and P(N) have the same cardinal. That is what Andreas responded to. I was suggesting that Woodin meant something else (as in fact I had taken him to), namely that each of the theories of these structures can be interpreted in the other.

Bill