Search results
Results from the WOW.Com Content Network
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.
Bernoulli assumed that "a poor man generally obtains more utility than a rich man from an equal gain" [3] an approach that is more profound than the simple mathematical expectation of money as it involves a law of moral expectation. Early theorists of utility considered that it had physically quantifiable attributes. They thought that utility ...
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 ...
Then (in the von Neumann formulation of cardinality) C is a set and therefore has a power set 2 C which, by Cantor's theorem, has cardinality strictly larger than C. Demonstrating a cardinality (namely that of 2 C) larger than C, which was assumed to be the greatest cardinal number, falsifies the definition of C. This contradiction establishes ...
Then an ordinal number is, by definition, a class consisting of all well-ordered sets of the same order type. To have the same order type is an equivalence relation on the class of well-ordered sets, and the ordinal numbers are the equivalence classes. Two sets of the same order type have the same cardinality.
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.
The proof of Cantor's second theorem came from Dedekind. However, it omits Dedekind's explanation of why the limits a ∞ and b ∞ exist. [45] Cantor restricted his first theorem to the set of real algebraic numbers. The proof he was using demonstrates the countability of the set of all algebraic numbers. [20]
In welfare economics and social choice theory, a social welfare function—also called a social ordering, ranking, utility, or choice function—is a function that ranks a set of social states by their desirability.