[FOM] Second order theories and Categoricity

[ali enayat]

This is a comment on the following query of Aatu Koskensilta (May 12, 2006):

>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?

Even without the restriction on axiomatizability, I doubt that one can 
produce a "nice" complete second order theory that is not categorical.

My hunch is partially based on a result of Victor Marek from the early 
1970's,  which shows that V=L implies that the second order theory of any 
countable structure S in a finite language determines the isomorphism type 
of S (Marek established a less general result, but his technique can be used 
to establish the aforementioned result, see below for more detail).

Perhaps one can even show that the above categorcity results holds for all 
Borel structures (under ZF + V=L).

Here is the reference for Marek's paper, and also Magidor's review of the 
paper:

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Marek, Wiktor, Consistance d'une hypothèse de Fraïssé sur la définissabilité 
dans un langage du second ordre.  C. R. Acad. Sci. Paris Sér. A-B 276 
(1973), A1147--A1150.

In these two papers the author proves the independence of Fraïssé's 
conjecture: If $\alpha$ and $\beta$ are two countable ordinals having the 
same second order theory, then $\alpha=\beta$. The author proves that 
Fraïssé's conjecture holds in the constructible universe. (In fact the proof 
can be slightly modified so as to show that in $V=L$ any two countable 
structures having a finite type and the same second order theory are 
isomorphic). On the other hand, the author remarks that in Azriel Lévy's 
model, in which the first inaccessible cardinal is collapsed to $\omega_1$, 
Fraïssé's conjecture fails.
(Reviewed by M. Magidor)
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Regards,

Ali Enayat