[FOM] An example of an axiomatizable second order theory that is complete but non-categorical?

[Robert M. Solovay]

Some further thoughts on Aatu's question:

 	1) In my last posting on this topic I considered only theories 
extending V= L + "my ordinals are a true" cardinal. I would now like to 
consider more general signatures and theories. I first continue to assume 
V=L. Because of the more general situation, it is necessary to have second 
order variables for each n which range over n-ary relations.

 	The following results are easily proved by the methods of my prior 
posting.

 	a) If A is an axiomitizable complete theory [in second order 
logic] then all its models have the same cardinality.

 	b) If A is finitely axiomitizable and complete, then A is 
categorical. {This uses ideas from the paper of Marek cited by Enayat.}

 	Even assuming V=L, I don't see how to prove every axiomitizable 
complete second order theory is categorical.

 	Next I want to present an example of a model where there is a 
finitely axiomitizable complete second order theory which *is not*
categorical.

 	Start with a countable transitve model, M, of ZFC + V=L. Add 
aleph_1 generic reals a la Cohen to the model getting a model N. [If it 
needs saying aleph_1 here is interpreted in the sense of the model M.]

 	Work inside N. The following theory is easily seen to be complete,
finitely axiomitizable, but not categorical.

 	The signature of A is that of ZFC.
A asserts: 1) ZF - power set {The two schemes are asserted inthe strong 
second order sense as single second order formulas.}

 	2) There is a largest cardinal and it is aleph_omega.

         3) A second order sentence saying that the order type of the 
ordinals is really a cardinal.

 	4) A second order sentence saying the ordinals of the model are 
well-ordered.

 	5) A first order sentence saying that the model is a Cohen 
extension of its constructibles obtained by adding a single generic real.

 	--Bob Solovay