TITLE: non-isomorphic objects with no known nontrivial distinguishing invariants
QUESTION [6 upvotes]: If X and Y are non-isomorphic objects, then "[is / is not] isomorphic to [ X / Y ]" are invariants that distinguish X and Y.  You can also do things like take an object Z that is not isomorphic to Y, and then "is isomorphic to X or isomorphic to Z" is another invariant that distinguishes X and Y.  Similarly, if W is isomorphic to X, then "is isomorphic to W" works too.

Are there any objects X and Y known to be non-isomorphic, but all known distinguishing invariants
(a) use the concept or definition of isomorphism

or

(b) use the concept or definition of being isomorphic to some object Z, where Z is isomorphic to X or Y

REPLY [4 votes]: Non-principal ultrafilters on the natural numbers don't have enough invariants to distinguish the non-isomorphic ones.  (Two such ultrafilters are called isomorphic if one is sent to the other by a permutation of the underlying set $\mathbb N$.)