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

[Aatu Koskensilta]

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?

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus