A social choice function is a function that takes as input the value-vector v and returns an outcome x \ in X.
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It can be stated by adding a new function symbol ? to the language of ZFC, with the property that ? is a global choice function.
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The meaning of the choice function guarantees that f ( i ) is in a _ i .-- Shahab 07 : 48, 2 October 2007 ( UTC)
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However, some countably infinite sets of nonempty sets can be proven to have a choice function in ZF without " any " form of the axiom of choice.
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In other words, the liberal paradox states that for every social choice function " F ", there is a configuration of preference relations " p"
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These are two different approaches to solving the issue; completeness is concerned with the input ( domain ) of the choice functions; while the strong axiom imposes conditions on the output.
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This means that it is possible to implement various non-linear social-choice functions, e . g . maximizing the sum-of-squares of values or the min-max value.
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A social choice function is a map which can take any configuration of preference relations of " N " as input and produce a subset of ( " chosen " ) social outcomes as output.
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According to the last phrase in Choice function # Bourbaki tau function it is mentioned by Hilbert in 1925 ( earlier than G�del 1940 ) . talk ) 17 : 54, 21 April 2013 ( UTC)
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Since the existence of a choice function when " X " is a finite set is easily proved from axioms "'1 8 "', AC only matters for certain infinite sets.
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