[FOM] Woodin's pair of articles on CH

[William Tait]

On Jan 14, 2010, at 12:11 AM, Monroe Eskew wrote:

> On Tue, Jan 12, 2010 at 3:29 PM, William Tait <williamtait at mac.com> wrote:
>> 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.
> 
> The problem is the "not necessarily uniquely" clause.  To produce the
> desired injection,  AC must be used to choose a tree relation on the
> naturals for each countable transitive set.

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