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

[Robert M. Solovay]

Aatu,

 	I think I see why no example of the sort you want exists.


 	I need for the argument that follows to assume V=L. Thus I leave 
the possibility open that in some models of set theory examples of the 
sort you want exist.

 	To pin things down I will consider the following sort of second order 
structure.

 	The underlying first order structure will be <L_kappa, epsilon> where 
kappa is an infinite cardinal in V. Since the first order structure has a 
pairing operation, we can take the second order variables to range just over 
subsets of L_kappa.

 	Let A be an axiomitizable theory in the obvious second order language 
which has at least one model of the sort described. I claim that if A is 
complete, it is categorical.

 	Suppose A is not categorical. Then we can add a new sentence theta to A 
that expresses the following: for no lambda < kappa which is a cardinal in V do 
all the sentences of A hold in the second order structure associated with 
L_lambda. [This uses the recursiveness of A and the fact that second order 
statements about L_lambda can be expressed in a first order way in L_kappa and 
hence in a uniform way in the second order version of L_kappa.]

 	A + theta is obviously consistent and categorical. Since A is not 
categorical, it is not complete.

 	--Bob Solovay


On Fri, 12 May 2006, Aatu Koskensilta wrote:

> Call a second order theory T complete if for every A either T |= A or T
> |= ~A. A simple cardinality argument shows that there are complete but
> non-categorical second order theories, but is there any nice example of
> an axiomatizable second order theory that is complete but
> non-categorical?
>