  This work presents theorems which state (i) Z is a proper subset for any
bijection f between A and Z, where Z is contained in P(A), A is a non-finite
set and |Z|=|A|, and (ii) being Z a proper subset of P(A) nothing affirms or
denies that |P(A)|>|A|. Russell's paradox is examined and it is shown that the
set of all the ordinary sets does not exist. A mistake in Cantor's proof on
cardinality of power sets is shown.
