TITLE: Categoricity in second order logic
QUESTION [10 upvotes]: Hi,
It's shown by an easy cardinality argument that there are complete second-order theories that are not categorical (have more than one model up to isomorphism). Anyone knows of a concrete example of such a theory?
Thanks in advance

REPLY [14 votes]: A theory of the type you are asking cannot be that concrete, because:


By an old result of Victor Marek, it is consistent with the axioms of $ZFC$ that the second order theory of every countable structure (in a countable vocabulary) is categorical. See this FOM-post of mine for a reference.
In the above FOM-post, I conjectured that Marek's result can be extended to all Borel structures. This conjecture was verified by Harvey Friedman in this FOM-post.
In yet another FOM-post, Solovay showed that it is consistent with $ZFC$ that as soon as a second order theory $T$ is both axiomatizable and complete, then $T$ is categorical. See also this other related FOM-post of Solovay.