Which, although too much to digest in a single read-through, perfectly segues into a crazy idea I had the other day.
Game Theory is a geometry, someone told me. The difficult part of voting math is often the cyclical nature of certain results (A>B B>C C>A) (anyone who reads this list can come to that conclusion quickly enough) Ergo... use a non-euclidian geometry. um, maybe I just meant... use a spherical geometry. Maybe arrow's theorem won't even apply with a limitation like "the voter space is finite" (like a sphere has a definite size, whereas a plane does not) or maybe i'm "mixing metaphors" (mixing mathematics) -JOSH NARINS (hypothetically proud) -----Original Message----- From: Forest Simmons [mailto:[EMAIL PROTECTED]] Sent: Wednesday, January 15, 2003 3:58 PM To: [EMAIL PROTECTED] Subject: Re: [EM] 1-Person-1-Vote has been abandoned. Actually, Cantor proved that there are infinitely many distinct infinities on the same day he proved that the cardinality of the reals is greater than the cardinality of the rationals. Here's the proof in modern notation: Let X be any set (finite or infinite, it doesn't matter). Let P(X) be the power set of X, i.e. the collection of all subsets of X. Then while there is a one-to-one function from X into P(X), there is no function from X onto P(X). In other words, the cardinality of P(X) is strictly greater than the cardinality of X. Let N be the set of natural numbers and (for any set A) let #A represent the cardinality of A. Then #N < #P(N) < #P(P(N)) < #P(P(P(N))), etc. Details: An example of a one-to-one function from X into P(X) is the function given by f(x)={x}, i.e. the image of any member of X is the singleton subset for which it is the only member. To show that no function maps X onto P(X), suppose to the contrary that g is such a function. Then let Y be the member of P(X) defined by { x | x is not a member of g(x)}. Since Y is a member of P(X), and g maps X onto P(X), there must be some member x of X such that g(x)=Y. Note that according to the definition of Y, this x is an element of Y if and only if it is not an element of Y. This contradiction shows the non-existence of any such map g. This is the famous diagonal argument of Cantor slightly disguised. Note that the points of the middle thirds Cantor set are in one to one correspondence with the base three "decimal" expansions that have no occurrence of the digit 1 . These expansions are, in one to one correspondence with subsets of N. For example x=.20020222202... corresponds to the subset h(x)={1,4,6,7,8,9,11,...}. In general h(x)={ n | the n_th digit of x is a two (in the base three expansion)}. So the Cantor set, which is a subset of the reals has the same cardinality as the power set of the natural numbers. Another of Cantor's ingenious arguments showed that the Naturals have the same cardinality as the Rationals. He was unable to answer the question of whether there might be other cardinalities between those of the Naturals and the Reals. Godel constructed a model of set theory in which there is no such cardinality. Later Cohen constructed a model in which there are infinitely many such cardinalities. So just as there are both Euclidean and non-Euclidean geometries, there are Cantorian and Non-Cantorian set theories. In other words, the "Continuum Hypothesis" is just as logically independent of the other basic axioms of set theory as the parallel postulate is independent from the other axioms of geometry. Forest On Wed, 15 Jan 2003, Eric Gorr wrote: > > I am assuming that no one has discovered another level of infinity. > ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em ------------------------------------------------------------------------------ This message is intended only for the personal and confidential use of the designated recipient(s) named above. If you are not the intended recipient of this message you are hereby notified that any review, dissemination, distribution or copying of this message is strictly prohibited. This communication is for information purposes only and should not be regarded as an offer to sell or as a solicitation of an offer to buy any financial product, an official confirmation of any transaction, or as an official statement of Lehman Brothers. Email transmission cannot be guaranteed to be secure or error-free. Therefore, we do not represent that this information is complete or accurate and it should not be relied upon as such. All information is subject to change without notice. ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
