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If s is in T, then by definition of T, s is not in f(s), so T is not equal to f(s). On the other hand, if s is not in T, then by definition of T, s is in f(s), so again T is not equal to f(s); cf. picture. For a more complete account of this proof, see Cantor's theorem.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers (see: (sequence A102288 in the OEIS)), and the set of all subsets of the set ...
As a consequence, the cardinality of the real numbers, which is the same as that of the power set of the integers, is strictly larger than the cardinality of the integers; see Cardinality of the continuum for details. The theorem is named for Georg Cantor, who first stated and proved it at the end of the 19th century.
A different meaning of cardinality was used by economists who followed the formulation of Hicks-Allen, where two cardinal utility functions are considered the same if they preserve preference orderings uniquely up to positive affine transformations.
Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers a < b , no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those ...
Sociology of sociology or metasociology is an area of sociology that combines social theories with analysis of the effect of socio-historical contexts in sociological intellectual production. [ citation needed ]
Sociology would be only marginally relevant to successful theories, which succeeded because they had revealed a fact of nature. The strong programme proposed that both "true" and "false" scientific theories should be treated the same way. Both are caused by social factors or conditions, such as cultural context and self-interest. All human ...
The proof is in two parts (Arrow, 1963, pp. 97–100). The first part considers the hypothetical case of some one voter's ordering that prevails ('is decisive') as to the social choice for some pair of social states no matter what that voter's preference for the pair, despite all other voters opposing.