By representing the social choice process as a " function " on " Rel ( X ) " " N ", we are tacitly assuming that the social choice function is defined for any possible configuration of preference relations; this is sometimes called the Universal Domain assumption .)
32.
Then our choice function can choose the least element of every set under our unusual ordering . " The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence; every set can be well-ordered if and only if the axiom of choice holds.
33.
Given a social choice function \ operatorname { Soc }, it is possible to build a social ranking function \ operatorname { Rank }, as follows : in order to decide whether a \ prec b, the \ operatorname { Rank } function creates new preferences in which a and b are moved to the top of all voters'preferences.
34.
A player i is called a "'dictator "'in a social-choice function \ operatorname { Soc } if \ operatorname { Soc } always selects the outcome that player i prefers over all other outcomes . \ operatorname { Soc } is called a "'dictatorship "'if there is a player i who is a dictator in it.
35.
It states that if a social choice function can be implemented by an arbitrary mechanism ( i . e . if that mechanism has an equilibrium outcome that corresponds to the outcome of the social choice function ), then the same function can be implemented by an incentive-compatible-direct-mechanism ( i . e . in which players truthfully report type ) with the same equilibrium outcome ( payoffs ).
36.
It states that if a social choice function can be implemented by an arbitrary mechanism ( i . e . if that mechanism has an equilibrium outcome that corresponds to the outcome of the social choice function ), then the same function can be implemented by an incentive-compatible-direct-mechanism ( i . e . in which players truthfully report type ) with the same equilibrium outcome ( payoffs ).
37.
:: : Not sure what you mean by a'set of an infinite real number of elements,'but certainly there are infinite sets with definable choice functions : for any ordinal the function'take the least element'is a well-defined choice function . btw, someone who knows how it's done should move this to the maths page Algebraist 15 : 39, 22 February 2007 ( UTC)
38.
:: : Not sure what you mean by a'set of an infinite real number of elements,'but certainly there are infinite sets with definable choice functions : for any ordinal the function'take the least element'is a well-defined choice function . btw, someone who knows how it's done should move this to the maths page Algebraist 15 : 39, 22 February 2007 ( UTC)
39.
If the method is applied to an infinite sequence ( " X " " i " : " i " " ? ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.
40.
If the method is applied to an infinite sequence ( " X " " i " : " i " " ? ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.
