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Yet another weird SF fan
 

Monday, February 18, 2008

Politics in an Infinite Society (or Ultrafilters and the Arrow Impossibility Theorem)

Ilkka Kokkarinen recently speculated on the nature of an infinite society (mentioned here). One possible effect is that if the society is actually infinite (not merely potentially infinite) and if it is somehow able to be politically unified (maybe it's the Republic of Heaven), then the Arrow Impossibility Theorem doesn't hold. According to a recent article in the January issue of the American Mathematical Monthly:

Suppose that in an election there are finitely many n(≤ 3) candidates {c1, … , cn} and a set X of voters. Each voter makes a ranking of the candidates, and the outcome of the election is determined by two rules:

  • if all the voters enter the same ranking, then this is the outcome;

  • whether a candidate a precedes candidate b in the outcome depends only on their order on the different ranking lists of the individual voters (and it does not depend on where a and b are on those lists; i.e., on how the voters ranked other candidates).

Show that there is an ultrafilter ℋ on X such that the outcome is an ordering π of {c1, … , cn} if and only if the set Fπ of those voters whose ranking is π belongs to ℋ.

In a finite set any ultrafilter is simply the set of all sets that include a specified point. When this is applied to politics, it means that there is a specified voter who acts as a dictator. (In the story “Franchise” by Isaac Asimov (loosely based on the 1952 Presidential election), such a dictator was compatible with the trappings of democracy.) In an infinite society, other ultrafilters are theoretically possible.

On the other hand, an infinite society would presumably have an infinite number of candidates. That would mean that the society would require a measurable cardinal number of voters.

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