Yet another weird SF fan


I'm a mathematician, a libertarian, and a science-fiction fan. Common sense? What's that?

Go to first entry


 

Archives

<< current
 
E-mail address:
jhertzli AT ix DOT netcom DOT com


My Earthlink/Netcom Site

My Tweets

My other blogs
Small Sample Watch
XBM Graphics


The Former Four Horsemen of the Ablogalypse:
Someone who used to be sane (formerly War)
Someone who used to be serious (formerly Plague)
Rally 'round the President (formerly Famine)
Dr. Yes (formerly Death)

Interesting weblogs:
Back Off Government!
Bad Science
Blogblivion
Boing Boing
Debunkers Discussion Forum
Deep Space Bombardment
Depleted Cranium
Dr. Boli’s Celebrated Magazine.
EconLog
Foreign Dispatches
Good Math, Bad Math
Greenie Watch
The Hand Of Munger
Howard Lovy's NanoBot
Hyscience
Liberty's Torch
The Long View
My sister's blog
Neo Warmonger
Next Big Future
Out of Step Jew
Overcoming Bias
The Passing Parade
Peter Watts Newscrawl
Physics Geek
Pictures of Math
Poor Medical Student
Prolifeguy's take
The Raving Theist
RealityCarnival
Respectful Insolence
Sedenion
Seriously Science
Shtetl-Optimized
Slate Star Codex
The Speculist
The Technoptimist
TJIC
Tools of Renewal
XBM Graphics
Zoe Brain

Other interesting web sites:
Aspies For Freedom
Crank Dot Net
Day By Day
Dihydrogen Monoxide - DHMO Homepage
Fourmilab
Jewish Pro-Life Foundation
Libertarians for Life
The Mad Revisionist
Piled Higher and Deeper
Science, Pseudoscience, and Irrationalism
Sustainability of Human Progress


























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.

0 Comments:

Post a Comment

<< Home

 
Profiles
My Blogger Profile
eXTReMe Tracker X-treme Tracker


The Atom Feed This page is powered by Blogger.