  A set theory is developed based on the approximations of sets and denoted by
AS. In AS the set of all sets exists but the argument for Russell's and
Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are
not valid. All the other axioms of ZF are valid and all the basic sets, such as
complement, intersection and cartesian product, exist although complement is
not quite the same set as in ZF. The set of all sets can be equipped with the
topology of approximations (Ta). Every set is closed and every function is
continuous in Ta. This implies that the Continuum Hypothesis is false. The sets
containing a subset which is perfect in Ta are of the greatest cardinality. A
simple observation shows that the concept of well-ordering must be defined in a
slightly different way than in ZF. We prove that a set can be well-ordered if
and only if it has no perfect subset. Therefore the cardinalities of arbitrary
sets are always comparable without assuming the Axiom of Choice. The cardinals
following the smallest infinite cardinal w are 2w, 3w, ..., w^2, 2w^2, 3w^2,
>..., w^3.... each being of greater cardinality than the previous one, which is
not the case in ZF. Immediately after these cardinals does not follow w^w which
is not a well-orderable set but some well-ordered cardinal k, and this one is
followed by the cardinals 2k, 3k, ..., k^2, 2k^2, 3k^2, ..., k^3...., etc. The
greatest cardinal is P(w) and is not a well-orderable set. The cofinality of a
well-ordered set is either 2 or w. The only regular cardinals are 0, 1, 2, w
and P(w). All other cardinals are singular. The only strong limit cardinal is
w. The only inaccessible cardinal is P(w). Strongly inaccessible cardinals do
not exist.
