![]() The set of real numbers, the continuum has the cardinality (also called continuum) The usual notation for the exponentiation of cardinalities is (9) Starting with any set A we may construct a sequence of sets (7)Īs a consequence of the theorem, the cardinalities of the sets in sequence (7) are all different. It follows that our assumption of the existence of a 1-1 correspondence between A and 2 A leads to a contradiction. But s ∉ S is also impossible, for if this were the case, s would be an element of S, by the same definition. It can't be that s ∈ S, because, by definition, S consists of those a for which a ∉ f(a). Indeed, either s belongs to S or it does not. Since f is assumed to be 1-1, it is surjective: there is an s ∈ A such that S = f(s). ![]() Let's define set S as a collection of the elements of a ∈ A that do not belong to their image f(a): (6) There are two possibilities: either a ∈ f(a) or not. Function f relates to each element a of A a subset f(a) of A. ![]() Suppose to the contrary that there is a 1-1 correspondence f: A → 2 A. So the term finite cardinal number is a synonym for natural number. Their common number of elements serves to denote their cardinality. Two finite sets have the same cardinality only if they have the same number of elements. Two sets are said to be of the same cardinality if there exists a 1-1 correspondence between the two. The key to a definition of cardinal numbers is the notion of a 1-1 correspondence.
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