Are two Grothendieck universes, such that one is not a subset of another, possible? It depends on the context: i suspect it is not provable in ZFC or in MK.(i use the second one for my purposes) The question is whether it is provably false.
The example would be good, but the real intent behind the question is "Is it possible to require a subset order on Grothendieck Universes"?
What will be a consequence of such demand? Contradiction or some stronger theory? Which one?
No. This isn't possible. Since every Grothendieck universe is essentially of the form $V_\kappa$ for some inaccessible cardinal $\kappa$, we get that if $U_0,U_1$ are two universes, there are two cardinals $\kappa_0,\kappa_1$ such that $U_i=V_{\kappa_i}$.
Since the cardinals are totally ordered (and in this case, $\kappa_i$ are ordinals, so choice is not even involved in their comparability), we get that $V_{\kappa_0}\subseteq V_{\kappa_1}$ or vice versa.