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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 the 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.

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An example of untrasportable relationsentence, 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 exist exists a bijection from X to the set Y, where not all elements of Y are finite sets.
show/hide this revision's text 1

An example of untrasportable relation, 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 exist a bijection from X to the set Y, where not all elements of Y are finite sets.