1:1 founctions that map the power set, P(N), onto a proper subset of N, which is the set containing all natural numbers, are disclosed. Thus, the cardinal number of P(N) is less than or equal to that of N. This appears to be contradictory to Cantor’s proof of cardinality. Cantor's proof was assuming the opposite first and then show it ended with a contradictory statement by deduction. However, this way of proof might be questionable since it's involved in an infinite domain and might not have a clear exclusive true or false choice.
Geeng-Chuan Chern (Sun,) studied this question.