TITLE: ultrapowers and higher order logic
QUESTION [12 upvotes]: One of the reasons I think ultrapowers are interesting is the following corollary of Łoś's theorem:

Let $V$ be a relational structure and $^*V$ an ultrapower of $V$.  Then a first order statement about $V$ is true if and only if the corresponding statement for $^*V$ is true, where the relations $R$ on $V$ are replaced by their enlargements $^*R$.

This suggests the following question:  what construction should replace the ultraproduct if I want a similar result for 2nd or higher order statements?  A little googling suggests that a book by van Benthem called Modal Logic and Classical Logic might contain an answer, but I don't have access to a copy.

REPLY [3 votes]: As pointed out by Matthew Towers, any ultrapower construction that preserves second order statements must be trivial at least in those cases where the original structure satisfies a categorical second order statement. 
But there may be fragments of second order logic that can be preserved by ultrapowers and/or ultraproducts.  

As a trivial example, any purely existential second order statement which is true in all factors will be true in the ultraproduct (since an expansion of the original structure by new predicates induces an expansion of the ultraproduct).  
As a trivial non-example: already the purely universal and monadic second order statement "every subset containing 0 and closed under successor is the whole set" is not preserved in any nontrivial ultrapower of the natural numbers, nor can it be preserved in any (nontrivial) proposed "advanced ultrapower construction".