[FOM] Woodin's pair of articles on CH
williamtait at mac.com
Fri Jan 15 08:26:08 EST 2010
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.
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