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



non-isomorphic objects with no known nontrivial distinguishing invariants
Hereditarily indecomposable Banach spaces are strange objects that fail to be isomorphic to any of their proper subspaces. However, in a certain sense those subspaces are not "interestingly different" from the spaces themselves. So there is no hope of finding an invariant to distinguish between the space and a subspace. The way the proof actually works is that one proves that every operator from the space to itself can be approximated, in a certain sense, by a multiple of the identity, and hence either has no hope of being an isomorphism or is Fredholm with index zero. But an isomorphism to a proper subspace would have to be Fredholm with positive (or infinite) index.


