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?



Bourbaki theory of isomorphism, examples of untransportable formulas
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. 


