The joke ran that only 6 people read all of Principia Mathematica, and three of them were liquidated by Hilter. This post is like that, almost no one but CSTAR and some of the other econogeeks should read it.
Or anyone who is trying to realize the basis of the conflict between liberalism, and such pseudo liberalisms as libertarianism. It's think and dense had has to do with Amartya Sen, Kenneth Arrow, John Rawls, Kurt Goedel and Georg Cantor.
It's important because it makes obsolete one of the big arguments in contemporary philosophy and critical theory, but that's for another day.
Without further ado, I bring you the constitution that Cantor and Goedel would write and the limits of rationality....
Amartya Sen is not the sort of person who makes many mistakes. This is why he is referred to, by almost anyone familiar with his work as "one of the great minds of the age." Thus, when one is reading a paper by Sen, and one notices a mistake - or more correctly, agreeing with someone else's mistake - it hits like a shot. I hope in passing that somehow these words will get to Micheal J. Sandel, because it has to with Rawls, and indeed his own work on Rawls. But to get to Sen's mistake requires a detour through Kenneth Arrow's General Possibility Theorem.
I Kurt and George, meet Ken, Ken, meet Kurt and George
The General Possibility Theorem states that given the conditions on a society of Pareto Optimality (P), Unbounded Choice (U), Independence of preferences (I) and Non-Dictatorship (D) there is no Social Choice function that will satisfy all of these in all cases.
This is true because it is another recurrance of Cantor's proof that the number of rational numbers is a set of a lower order than the number of the continuum.
Proof
Let c be the set of choices, so long as it is finite. Let u be the set of untility functions in the society. By U and I, for any individual i, u(c)>>s which is a finite string which describes the ordering of choices of i. Not all preferences need to be strict, since : can be strict ordering where as ; can be indifference, and by I, there is no effect of substituting ; for : in any pair - otherwise the pairings aren't independent.
Let v be the proposed social choice function to be applied to the set of all s orderings. v( {s})>> d where d is the decision. But d is also an s, that is it contains the same symbols as the string s, and therefore is a properly constructed s.
Thus for each individual i, d is a member of c, that is a choice, and therefore, by U there must be an ordering of all possible d's, and they are independent of each other. Thus {s} must contain {d} and v({s}) must apply to {d}, and therefore u({d}) is also a properly constructed string.
For v({s}) to produce a d then, all v({s}) must terminate. That is, there must be a one to one correspondance, at most, with the set of rational numbers, because the set of rational numbers is the set of all numbers which can be expressed by finite strings with a finite number of symbols. Since we have asserted that c,s,d are finite, and there are a finite number of operations which express their orderings, this is so.
Thus for each i, their u(s) implies some set of possible v({s)). If the voting system is not null, then v({s}) must yield a set of possible results p{d} which is not the same as {d}. That is, if a particular vote does not have any chance of changing the possible outcomes, it is not a vote. If all voters have no chance of changing the outcome, then D is violated, and if only one has the chance of changing the outcome, then D is violated. So for D not to be violated for any v({s}) there must be at least one i for which altering his u(s) will improve u(d). Because if he were indifferent, then his vote would not matter, and we would be back to a violation of D.
Since u(c) produces an s, and v({s}) produces a d, and d is a properly formed c, then u(c) is also a properly formed string for choice, because it implies a properly formed s. Therefore when faced with the possibility of changing u(c) to produce a potentially better d, there will be at least one vote who will change their u, to produce a better result set d.
But once he has done this, by P, some other individual must also be worse off, otherwise his vote doesn't count, and he will make the same decision to change his u to produce a better possible result set {d} that is one where the new d is preferred to the old d.
Thus for any ordering of the set of decisions strings s when mapped to the set of decision strings d, there is always one v(d) which can be changed to require a new mapping. Thus, v(d) does not terminate, and is therefore of a higher order than the number of rational numbers. QED.
This can be proved in any symbol system any one cares to, because U,P,I,D are a calculus C which is rich enough to encompass all of Arithmatic, and by Goedel, must have undecideable propositions in it. One can then chase ones tail about how to decide the undecidable propositions and find that one must violate one of one's axioms. The string proof, that is the equivalence of finite strings to rational numbers, and the proofs to irrational pops out of this as well.
In short, there is no way "out" of Arrow's theorem, because it is simply equivalent in econospeak to Cantor's theory of infinite sets and Goedel's realization that the set of proofs must be the set of Rational numbers, where as the set of truths is the set of Irrational numbers.
Thus the minimal problem can be phrased this way: if there will always be a final choice, and everyone's input is important in that final choice, and everyone can understand the consequences of their choice then there must be a point where individuals must be prevented from changing their choice. The minimum paradox is then Finality, Understanding and Input. In short, rationality is insufficient to make choices, so long as we must understand what we do, come to an end of all decisions, and everyone has at least some input.
Academia is rent, the problem is not the proof, but proving that no one else has proven the proof. Since people are still disputing Arrow as of 1997 (by Google), I'm going to assert that no one has come up with this and communicated it, even though it's obvious, which means someone has. So who ever you are, publish damn it.
What this means is that the recent developments in logic, including proof of Fermat, have social choice equivalents. In fact, for any set of axioms, there is an equivalent calculus, and the soical choice questions - what can be decided by it - can have all of the mathematical tools developed in the last century applied to it.
A Rawling Metropolis
As Clarke said of Einstien "he can't be defined, but he can be evaded". One can produce subset calculuses which do not have the property of resulting in truths that are of order of the continuum. One way is to violate independence - that is, force trade offs between different orders. This is the road of everyone who argues that democracy should be subordinate to the market mechanism. If there is a cost for different orderings, then eventually some i will run out of money to "change" his vote, and the entire process terminates.
Sen proved however, that this means a restriction on minimal liberty, and therefore by his "Liberal Paradox" it is impossible to be a Paretan liberal. Either some amount of the set must be burned off enforcing the decision, in which case it isn't Pareto optimal, or someone must sacrifice some of their liberty.
Another way to evade this is to say that suppose not all choices need to have a strict order. Let's call this <|x,y|> and express the result in a string as state q that is x q y means that in a given set s, it is not known until there is a d whether x or y is prefered, but one will be. What this says is that a given i does not have enough information to choose betwen x and y before hand, but when x or y are chosen, then there will be some preference x:y, y:x, x;y.
It doesn't help that one may have an I whose vote matters who doesn't know how it matters, because the fact that he doesn't know is different from his knowing and having a preference. Either one must say that q means that his decision doesn't matter, or one must say that he is forced to order q before he has the information to make a choice. This is equivalent to the bomb problem detailed by Penrose - if the choice could matter, then the result is different than if it can't matter.
Would this get out of the paradox? No, because even if we substitute q into the strings, there are still preferred strings left, by I u(p) must yield a string,and therefore we are back where we started.
Unless all orderings are q for all i.
This is equivalent to the Rawlsian veil of ignorance (which Sen noticed, even though he did not realize that there was a quantum formalism to express it). If there is no way of anyone knowing whether they will prefer the results, they will finally give up enough freedom of choices to gravitate to the maximally egalitarian pareto optimum. Now there is a proof that shows that this state, let's call it the Rawls Equilibrium, will have a unique property - among i who are completely self-centered and have the same u with respect to themselves, the RE will be stable in the rankings. One can form a social choice function in a society of devils, as Madison sensed, by taking the most stable choice out of all of them.
This also attaches all of the questions of imperfect information to quantum mathematics, for a given insoluable social choice problem, one can find out which pieces of information can be withheld from whom to produce a final choice, or which pieces of information can be given to all to produce the maximum equilibrium.
We can finally get to where Sen gets shafted by his own Arrow. Or rather where he agrees with James Buchanan in error. In his book Rationality and Freedom, Sen collects several of his papers on social choice - including responses to his own liberal paradox and Arrow's GPT. He even mentions Kurt Goedel, but in passing. Therefore I am fairly sure that if someone has published the equivalence of the FU and the GPT, it wasn't in Sen's knowledge in 2002.
He also remarks in passing that James Buchanan notes that P,I,D,U represent the "constitution" of the social choice.
This is wrong. The social choice function S represents the constitution, P, I, D and U represent one configuration of what liberal theory has, for some centuries now, referred to as "natural law". And their failure is when that "natural law" is combined with the finite nature of human beings. If gods of infinite duration and no needs played with different social choices, they could come to a resolution at any time they decided that taking the current decision brings greater happiness than continuing to decide. That is, as long as there is no down side that really hurts, why not give way.
In reality, since v is a properly constructed string, and if f represents {v} which can be selected, then u(f) must be an order of constitutions, or social choices. And this is the practical way out of Arrow - one that Sen hints at in the introduction to Rationality and Freedom but which has a mathematical equivalent: namely, if there is a "market of markets", then which social choices are no longer fixed - people will flow towards the market that gives them the best chance of getting the final d they want, and then flow away as that {d} becomes clearer. The quantum choices are resolved, and this becomes a new decision.
As long as people can stay in the game.
This then is the mathematical root of liberalism versus libertarianism and all forms of property dominance. As long as everyone can stay in the game, the soical choice can continue to seek justice and pareto optimality, but when people can be taken out, by the results of the game, they will either be forced out, and be suboptimal, or they will quit, and be suboptimal.
The constitutional process is not the principles of P, U, I and D - these are presumptions that are beyond a vote, and are in place, not because they are asserted by any individual, but because if they are violated, there will be individuals out of the game, who will then change whatever v has been selected and assert a new one. But that's for another day.
Also for another day is to point out that much of late 20th century thinking is an attempt to solve special cases of this problem by stretching one of the initial assumptions. For example quantum mechanics has discrete particles, this yields a contradiction, and so string theory says - well what happens if our truth points aren't points, but are continuous? What if we do logic with continous entities rather than string point entities?