TITLE: Bourbaki theory of isomorphism, examples of untransportable formulas
QUESTION [5 upvotes]: In their book "Theory of sets" Bourbaki suggested a general theory of isomorphism.
(See also http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf )
The example of an untransportable relation (i.e. formula) in the book involves 2 principal base sets.
Are there examples of untrasportable formulas when we allow only one principal base set?

REPLY [3 votes]: An example of untrasportable sentence, when there is only one principal base set X, may be the following one: 
All elements of the set X are finite sets,
Because, by definition, the truth value of a transportable sentence must be preserved under all bijections from the set X. Obviously, there  exists a bijection from X to a set Y, where not all elements of Y are finite sets.
A simpler example is "the set X contains the empty set". 
There is a paper "Sentences of type theory: the only sentences preserved under isomorphisms" by Victoria Marshall and Rolando Chuaqui - see The Journal of symbolic Logic, vol 56, #3, Sep 1991.