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A set theory is developed based on the approximations of sets and denoted byAS. In AS the set of all sets exists but the argument for Russell's andCantor's paradox fail. The Axioms of Separation, Replacement and Foundation arenot valid. All the other axioms of ZF are valid and all the basic sets, such ascomplement, intersection and cartesian product, exist although complement isnot quite the same set as in ZF. The set of all sets can be equipped with thetopology of approximations (Ta). Every set is closed and every function iscontinuous in Ta. This implies that the Continuum Hypothesis is false. The setscontaining a subset which is perfect in Ta are of the greatest cardinality. Asimple observation shows that the concept of well-ordering must be defined in aslightly different way than in ZF. We prove that a set can be well-ordered ifand only if it has no perfect subset. Therefore the cardinalities of arbitrarysets are always comparable without assuming the Axiom of Choice. The cardinalsfollowing 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 isnot the case in ZF. Immediately after these cardinals does not follow w^w whichis not a well-orderable set but some well-ordered cardinal k, and this one isfollowed by the cardinals 2k, 3k, ..., k^2, 2k^2, 3k^2, ..., k^3...., etc. Thegreatest cardinal is P(w) and is not a well-orderable set. The cofinality of awell-ordered set is either 2 or w. The only regular cardinals are 0, 1, 2, wand P(w). All other cardinals are singular. The only strong limit cardinal isw. The only inaccessible cardinal is P(w). Strongly inaccessible cardinals donot exist.

Disproof of the Continuum Hypothesis and Determination of the Cardinality of Continuum by Approximations of Sets