----- Original Message ---- From: Fred lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Saturday, October 27, 2007 12:44:41 PM Subject: Re: [math-fun] Re: radio math On 10/27/07, Henry Baker <hbaker1@pipeline.com> wrote:
Cantor-type diagonalization is the major tool Goedel's work and in
computational complexity, so Cantor's work does find a role in certain parts of computer science.
It's an interesting challenge to figure out what role diagonalization
might play in "real" life. I'm not entirely convinced that the similarity of the Cantor diagonal argument to the Russell paradox / Goedel proof / Cretan liar et al is sufficient grounds for drawing the latter into a discussion about infinities. Perhaps it might be argued that diagonalisation leads to a new object in an open-ended system (e.g. the continuum in the cardinals); but in a closed system it leads to a contradiction. At any rate, I am convinced that diagonalisation by itself does impinge very immediately on "real life", in a manner which does not seem well appreciated. There is an obscure corner of mathematical logic and computer science known as "situation theory" [one reason for its low profile may well be that none of its practitioners will actually admit to being a situation theorist!]. Situation theory employs axiomatic set theory to model the concept of truth in a closed universe where the active agents are capable of self-reference. The central revelation is that (loosely) in such situation, either there is are questions whose answers are unknown to some agents, or else there are answers about which agents must disagree. There is a very readable account of all this material in Jon Barwise & John Etchemendy's book "The Liar". Failure to appreciate the ramfiications of this theorem have resulted elsewhere in the generation of an enormous amount of otherwise (possibly) well-informed and entertaining twaddle --- a situation which will doubtless persist, whatever I say! Fred, This sounds interesting. Can you give us an example of how situation theory impacts ordinary life? Gene __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com