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

Robert M. Solovay solovay at Math.Berkeley.EDU
Mon May 15 00:54:35 EDT 2006

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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?
>
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