[FOM] Second order theories and Categoricity

[Harvey Friedman]

On 5/13/06 3:05 PM, "ali enayat" <a_enayat at hotmail.com> wrote:

> Perhaps one can even show that the above categorcity results holds for all
> Borel structures (under ZF + V=L).
> 
How about this try. Argue in ZFC + V = L.

Let M be a Borel structure, with domain P(omega). We assume that the
relational type is countable.

Choose lambda to be the least limit ordinal such that the following holds.
If any two Borel structures are isomorphic, then there is an isomorphism in
L(lambda). 

It is not hard to see that the structure (L(lambda),epsilon) is second order
categorical. In fact, there is a single sentence phi of second order logic
such that (L(lambda),epsilon) is the unique model.

Now, in (L(lambda),epsilon), we can, given any Borel structure, define its L
least Borel code in a first order way.

Fix x containedin omega to be the L least Borel code for M.

We now can form the set of second order sentences that determine M up to
isomorphism. 

Fix n in x. We ask that there exists S containedin P(omega)^2 and u,v in
P(omega), such that

1. (P(omega),S) satisfies phi.

2. The point u is satisfied in (P(omega),S) to be a Borel structure whose L
least Borel code is v. This is all stated within (P(omega),S).

3. n lies in v, in the sense of (P(omega),S).

4. There is an external isomorphism between M and the internal structure u.

Fix n not in x. We ask that there exists S containedin P(omega)^2 and u,v in
P(omega), such that

1. (P(omega),S) satisfies phi.

2. The point u is satisfied in (P(omega),S) to be a Borel structure whose L
least Borel code is v. This is all within (P(omega),S).

3. n does not lie in v, in the sense of (P(omega),S).

4. There is an external isomorphism between M and the internal structure u.

Do you believe this?

Harvey