It is known that in this chain of inclusions:
$$ P \subseteq NP \subseteq PSPACE \subseteq EXPTIME $$ at least one inclusion is proper (ie, equality does not hold), by the "Time Hierarchy theorem". Other than that, none of the inclusions have been proven to be proper so far. Note that this theorem is proved via diagonalization, ie, Cantor's theorem that the cardinality of a power set must be bigger than the cardinality of the set itself. In other words, the proof is based on cardinality. But (with my limited knowledge and saying this vaguely) I have not known of similar cardinality proofs that are "smaller" than Cantor's. Does it mean that there is only 1 inequality within that chain?
