Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Robert Low on 29 Jul 2005 14:18 Han de Bruijn wrote: > Robert Low wrote: >> What you're proposing doesn't refine set theory, it >> breaks it into tiny little pieces, and then jumps >> up and down on the tiny little pieces until they're >> broken too. > I wonder what you think of my "Boolean Objects": I wonder why you didn't reply to any of the stuff preceding that paragraph: you know, the stuff explaining some of the objections to a={a}?
From: Robert Kolker on 29 Jul 2005 14:36 malbrain(a)yahoo.com wrote: > > Actually, Virgil, coloring trees and their parts is a useful concept in > graph theory. karl m "coloring" trees is simply applying labels. It has nothing to do inherently with frequency of electromagnetic radiation. The colors used in graph theory are simply conventional labels. Bob Kolker >
From: malbrain on 29 Jul 2005 16:57 david petry wrote: > I'm in the process of writing an article about > objections to Cantor's Theory, which I plan to contribute > to the Wikipedia. I would be interested in having > some intelligent feedback. Here' the article so far. > > *** > > While the pure mathematicians almost unanimously accept > Cantor's Theory (with the exception of a small group of > constructivists), there are lots of intelligent people who > believe it to be an absurdity. Typically, these people > are non-experts in pure mathematics, but they are people > who have who have found mathematics to be of great practical > value in science and technology, and who like to view > mathematics itself as a science. What research did you do to determine the size of the GROUPS in question? Did you enumerate them, or are you drawing analogies? > These "anti-Cantorians" see an underlying reality to > mathematics, namely, computation. They tend to accept the > idea that the computer can be thought of as a microscope > into the world of computation, and mathematics is the > science which studies the phenomena observed through that > microscope. They claim that that paradigm includes all > of the mathematics which has the potential to be applied to > the task of understanding phenomena in the real world (e.g. > in science and engineering). No, we're interested in CHANGING the real world. > Cantor's Theory, if taken seriously, would lead us to believe > that while the collection of all objects in the world of > computation is a countable set, and while the collection of all > identifiable abstractions derived from the world of computation > is a countable set, Bzzzt. There's no limit to the number of stories that can be told. karl m
From: Dave Rusin on 29 Jul 2005 21:25 In article <b779a$42e9e8d1$82a1e3ad$17135(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >Under the new rules {{}} = {} . Oh, that's just great. Tell me, what is the cardinality of the power set of, say {1,2} ? I would say there are four elements: P(S) = { {1,2}, {1}, {2}, {} } Would you now say there are just three? Which of these statements is true? P(S) = { {1,2}, {1}, {2} } union { {} } P(S) = { {1,2}, {1}, {2} } union {} | A union B | + | A intersect B | = |A| + |B| for all sets A,B . See, it's fine to believe that mathematicians are playing useless games. The problem is, they hold themselves to a high standard of consistency. You're welcome to introduce new rules. But you must either face the consequences of those rules or else spend your life inventing new ad-hoc rules every time your previous changes force you to a contradiction. Of course, mathematicians are human[*] and make mistakes but in general you can be sure that if they claim something is true, it is indeed a necessary consequence of their starting axioms, which are usually very few and certainly open to inspection. If you don't like the conclusions, you need to see which of the _axioms_ you find needs to be changed. (You are of course free to change those axioms and discover the consequences of your new set of axioms.) By the way, your claim elsewhere that an empty box is the same as "nothing" shows that you have not dealt with people in the trades. If the floor-polisher specifies in his contract that the room be empty, you will pay extra if you leave an empty box on the floor. dave [*] Not me, I'm a dog. This is the Internet.
From: malbrain on 30 Jul 2005 00:47
Virgil wrote: > TO says above, and I quote, "Any subset of N has a size that is in N." > So TO raises the issue of whether certain "sizes" are in N or not. Have you received the polemic on this yet? > As far as I can see, PJ is the first and only person to mention color in > connection with properties of sets. So perhaps PJ should be the one to > answer his own questions, seeing that he seems to be the only one whom > they interest. Is Mr Joker satisfied with the outcome of his NOMINATION? You got a hearty I CONCUR from Mr Virgil. karl m |