On Cantor's important theorems

Cantor proved that the n−dimensional manyfold remains continuous if the set of points with purely algebraic coordinates is removed. This proof is simplified and it is shown, that continuity is also preserved if the set of points with purely transcendental coordinates is removed. Cantor’s first proof of the uncountability of real numbers is modified to cover the set of algebraic numbers and even the set of rational numbers too. The principles and limitations of Cantor’s famous second diagonalization method are made easier comprehensible by simplifying Cantor’s infinite list. It facilitates to grasp the relation between natural numbers and rational numbers as well as between natural numbers and their powerset. The contradiction of any bijection between a set and its powerset can also be interpreted as evidence against the existence of any infinite sets, implying that transfinite cardinalities are meaningless. Some questions on the qualitative distinction between ℵ0 and 2^�ℵ0 � and the magnitude of n for logn approaching infinity support this point of view.